Tunable acoustic gradient index of refraction lens and system

ABSTRACT

A tunable acoustic gradient index of refraction (TAG) lens and system are provided that permit, in one aspect, dynamic selection of the lens output, including dynamic focusing and imaging. The system may include a TAG lens and at least one of a source and a detector of electromagnetic radiation. A controller may be provided in electrical communication with the lens and at least one of the source and detector and may be configured to provide a driving signal to control the index of refraction and to provide a synchronizing signal to time at least one of the source and the detector relative to the driving signal. Thus, the controller is able to specify that the source irradiates the lens (or detector detects the lens output) when a desired refractive index distribution is present within the lens, e.g. when a desired lens output is present.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.14/039,741, filed Sep. 27, 2013, which is a continuation of U.S. patentapplication Ser. No. 13/473,364, filed May 16, 2012, which is acontinuation of U.S. patent application Ser. No. 12/528,347, filed Mar.11, 2010, now issued as U.S. Pat. No. 8,194,307, which is a nationalstage entry of International Application No. PCT/US08/54892, filed Feb.25, 2008, which claims the benefit of priority of U.S. ProvisionalApplication Nos. 60/903,492 and 60/998,427, filed on Feb. 26, 2007 andOct. 10, 2007, respectively, the entire contents of which application(s)are incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates generally to a tunable acoustic gradientindex of refraction (TAG) lens, and more particularly, but notexclusively, to a TAG lens that is configured to permit dynamic focusingand imaging.

BACKGROUND OF THE INVENTION

When it comes to shaping the intensity patterns, wavefronts of light, orposition of an image plane or focus, fixed lenses are convenient, butoften the need for frequent reshaping requires adaptive opticalelements. Nonetheless, people typically settle for whatever comes out oftheir laser, be it Gaussian or top hat, and use fixed lenses to producea beam with the desired characteristics. In laser micromachining, forinstance, a microscope objective will provide a sharply focused regionof given area that provides sufficient power density to ablate thematerials.

However, in a variety of applications, it is useful or even necessary tohave feedback between the beam properties of the incident light and thematerials processes that are induced. A classical example is using atelescope to image distant objects through the atmosphere. In this case,the motion of the atmosphere causes constant perturbations in thewavefront of the light. One can measure the fluctuations and, usingadaptive elements, adjust the wavefront to cancel out these effects.Still other, laboratory-based, imaging applications such asophthalmologic scanning, confocal microscopy or multiphoton microscopyon living cells or tissue, would benefit greatly from the use of directfeedback to correct for wavefront aberrations induced by the sampleunder investigation, or to provide rapid scanning through focal planes.

Advanced materials processing applications also require precise beamintensity or wavefront profiles. In these cases, unlike imaging, one ismodifying the properties of a material using the laser. For instance,laser forward-transfer techniques such as direct-write printing candeposit complex patterns of materials—such as metal oxides for energystorage or even living cells for tissue engineering—onto substrates. Inthis technique, a focused laser irradiates and propels a dropletcomposed of a mixture of a liquid and the material of interest toward anearby substrate. The shape of the intensity profile of the incidentlaser plays a critical role in determining the properties of thedeposited materials or the health of a transferred cell. In cases suchas these, the ability to modify the shape of the incident beam isimportant, and with the ability to rapidly change the shape, one addsincreased functionality by varying the laser-induced changes in amaterial from one spot to another.

Even traditional laser processes like welding or cutting can benefitfrom adaptive optical elements. In welding, a continuous-wave lasermoves over a surface to create a weld bead between the two materials.Industrial reliability requires uniform weld beads, but slightfluctuations in the laser, the material, or the thermal profile candiminish uniformity. Therefore, with feedback to an adaptive opticalelement, more consistent and regular features are possible.

Whether the purpose is to process material, or simply to create animage, the applications for adaptive optics are quite varied. Somerequire continuous-wave light, others need pulsed light, but theunifying requirement of all applications is to have detailed controlover the properties of the light, and to be able to change thoseproperties rapidly so that the overall process can be optimized.

Fixed optical elements give great choice in selecting the wavefrontproperties of a beam of light, but there exist few techniques formodifying the beam temporally. The simplest approach is to mount a lensor a series of lenses on motion control stages. Then one can physicallytranslate the elements to deflect or defocus the beam. For instance,this technique is useful for changing the focus of a beam in order tomaintain imaging over a rough surface, or changing the spot size of afocused beam on a surface for laser micromachining. However, thisapproach suffers from a drawbacks related to large scale motion such asvibrations, repeatability and resolution. Moreover, it can be slow andinconvenient for many industrial applications where high reliability andspeed are needed. Nonetheless, for certain applications such as zoomlenses on security cameras, this is a satisfactory technique. Recently,more advanced methods of inducing mechanical changes to lenses involveelectric fields or pressure gradients on fluids and liquid crystals toslowly vary the shape of an element, thereby affecting its focal length.

When most people think of adaptive optical elements, they think of twocategories, digital micromirror arrays and spatial light modulators. Adigital mirror array is an array of small moveable mirrors that can beindividually addressed, usually fabricated with conventional MEMStechniques. The category also includes large, single-surface mirrorswhose surfaces can be modified with an array of actuators beneath thesurface. In either case, by controlling the angle of the reflectingsurfaces, these devices modulate the wavefront and shape of lightreflected from them. Originally digital mirror arrays had only twopositions for each mirror, but newer designs deliver a range of motionand angles.

Spatial light modulators also modify the wavefront of light incident onthem, but they typically rely on an addressable array of liquid crystalmaterial whose transmission or phase shift varies with electric field oneach pixel.

Both digital mirror arrays and spatial light modulators have broadcapabilities for modulating a beam of light and thereby providingadaptive optical control. These are digitally technologies and cantherefore faithfully reproduce arbitrary computer generated patterns,subject only to the pixilation limitations. These devices have gainedwidespread use in many commercial imaging and projecting technologies.For instance, digital mirror arrays are commonly used in astronomicalapplications, and spatial light modulators have made a great impact onprojection television and other display technology. On the researchfront, these devices have enabled a myriad of new experiments relying ona shaped or changeable spatial pattern such as in optical manipulation,or holography.

Although current adaptive optical technologies have been successful inmany applications, they suffer from limitations that prevent their useunder more extreme conditions. For instance, one of the majorlimitations of spatial light modulators is the slow switching speed,typically on the order of only 50-100 Hz. Digital mirror arrays can befaster, but their cost can be prohibitive. Also, while these devices aregood for small scale applications, larger scale devices require eitherlarger pixels, leading to pixilation errors, or they require anuntenable number of pixels to cover the area, decreasing the overallspeed and significantly increasing the cost. Finally, these devices tendto have relatively low damage thresholds, making them suitable forimaging applications, but less suitable for high energy/high power laserprocessing. Accordingly, there is a need the in the field of adaptiveoptics for devices which can overcome current device limitations, suchas speed and energy throughput for materials processing applications.

SUMMARY OF THE INVENTION

To overcome some of the aforementioned limitations, the presentinvention provides an adaptive-optical element termed by the inventorsas a “tunable-acoustic-gradient index-of-refraction lens”, or simply a“TAG lens.” In one exemplary configuration, the present inventionprovides a tunable acoustic gradient index of refraction lens comprisinga casing having a cavity disposed therein for receiving a refractivematerial capable of changing its refractive index in response toapplication of an acoustic wave thereto. To permit electricalcommunication with the interior of the casing, the casing may have anelectrical feedthrough port in the casing wall that communicates withthe cavity. A piezoelectric element may be provided within the casing inacoustic communication with the cavity for delivering an acoustic waveto the cavity to alter the refractive index of the refractive material.In the case where the refractive material is a fluid, the casing mayinclude a fluid port in the casing wall in fluid communication with thecavity to permit introduction of a refractive fluid into the cavity.Additionally, the casing may comprise an outer casing having a chamberdisposed therein and an inner casing disposed within the chamber of theouter casing, with the cavity disposed within the inner casing and withthe piezoelectric element is disposed within the cavity.

In one exemplary configuration the piezoelectric element may comprise acylindrical piezoelectric tube for receiving the refractive materialtherein. The piezoelectric tube may include an inner cylindrical surfaceand an outer cylindrical surface. An inner electrode may be disposed onthe inner cylindrical surface, and the inner electrode may be wrappedfrom the inner cylindrical surface to the outer cylindrical surface toprovide an annular electric contact region for the inner electrode onthe outer cylindrical surface. In another exemplary configuration, thepiezoelectric element may comprise a first and a second planarpiezoelectric element. The first and second planar piezoelectricelements may be disposed orthogonal to one another in an orientation forproviding the cavity with a rectangular cross-sectional shape.

The casing may comprise an optically transparent window disposed atopposing ends of the casing. At least one of the windows may include acurved surface and may have optical power. One or more of the windowsmay also operate as a filter or diffracting element or may be partiallymirrored.

In another of its aspects, the present invention provides a tunableacoustic gradient index of refraction optical system. The optical systemmay include a tunable acoustic gradient index of refraction lens and atleast one of a source of electromagnetic radiation and a detector ofelectromagnetic radiation. A controller may be provided in electricalcommunication with the tunable acoustic gradient index of refractionlens and at least one of the source and the detector. The controller maybe configured to provide a driving signal to control the index ofrefraction of the lens. The controller may also be configured to providea synchronizing signal to time at least one of the emission ofelectromagnetic radiation from the source or the detection ofelectromagnetic radiation by the detector relative to the electricalsignal controlling the lens. In so doing, the controller is able tospecify that the source irradiates the lens (or detector detects thelens output) when a desired refractive index distribution is presentwithin the lens. In this regard, the source may include a shutterelectrically connected to the controller (or detector) for receiving thesynchronizing signal to time the emission of radiation from the source(or detector).

The controller may be configured to provide a driving signal that causesthe focal length of the lens to vary with time to produce a lens with aplurality of focal lengths. In addition, the controller may beconfigured to provide a synchronizing signal to time at least one of theemission of electromagnetic radiation from the source or the detectionof electromagnetic radiation by the detector to coincide with a specificfocal length of the lens. In another exemplary configuration, thecontroller may be configured to provide a driving signal that causes thelens to operate as at least one of a converging lens and a diverginglens. Likewise the controller may be configured to provide a drivingsignal that causes the lens to operate to produce a Bessel beam outputor a multiscale Bessel beam output. Still further, the controller may beconfigured to provide a driving signal that causes the optical output ofthe lens to vary with time to produce an output that comprises a spot atone instance in time and an annular ring at another instance in time. Insuch a case, the controller may be configured to provide a synchronizingsignal to time at least one of the emission of electromagnetic radiationfrom the source or the detection of electromagnetic radiation by thedetector to coincide with either the spot or the annular ring outputfrom the lens. As a further example, the controller may be configured toprovide a driving signal that causes the optical output of the lens tovary with time to produce an output that comprises a phase mask or anarray of spots. To facilitate the latter, the lens may comprise arectangular or square cross-sectional shape.

As a still further exemplary configuration the controller may beconfigured to provide a driving signal that creates a substantiallyparabolic refractive index distribution, where the refractive index inthe lens varies as the square of the radius of the lens. Thesubstantially parabolic refractive index distribution may existsubstantially over the clear aperture of the lens or a portion of theaperture. In turn, the source of electromagnetic radiation may emit abeam of electromagnetic radiation having a width substantially matchedto the portion of the clear aperture over which the refractive indexdistribution is substantially parabolic. In this regard the source mayinclude an aperture to define the width of the emitted beam.Alternatively, the controller may be configured to provide a drivingsignal that creates a plurality of substantially parabolic refractiveindex distributions within the lens. The driving signal may comprise asinusoid, the sum of at least two sinusoidal driving signals ofdiffering frequency and/or phase, or may comprise a waveform other thana single frequency sinusoid.

In another of its aspects, the present invention provides a method fordriving a tunable acoustic gradient index of refraction lens to producea desired refractive index distribution within the lens. The methodincludes selecting a desired refractive index distribution to beproduced within the lens, determining the frequency response of thelens, and using the frequency response to determine a transfer functionof the lens to relate the index response to voltage input. In additionthe method includes decomposing the desired refractive indexdistribution into its spatial frequencies, and converting the spatialfrequencies into temporal frequencies representing the voltage input asan expansion having voltage coefficients. The method further includesdetermining the voltage coefficients from the representation of thedecomposed refractive index distribution, and using the determinedvoltage coefficients to determine the voltage input in the time domain.The method then includes driving a tunable acoustic gradient index ofrefraction lens with the determined voltage input. In this method, thedecomposed refractive index distribution may be converted into discretespatial frequencies to provide a discretized representation of thedecomposed refractive index distribution.

In yet another its aspects, the invention provides a method forcontrolling the output of a tunable acoustic gradient index ofrefraction optical lens. The method includes providing a tunableacoustic gradient index of refraction lens having a refractive indexthat varies in response to an applied electrical driving signal, andirradiating the optical input of the lens with a source ofelectromagnetic radiation. In addition the method includes driving thelens with a driving signal to control the index of refraction within thelens, and detecting the electromagnetic radiation output from the drivenlens with a detector. The method then includes providing a synchronizingsignal to the detector to select a time to detect the electromagneticradiation output from the driven lens when a desired refractive indexdistribution is present within the lens.

In still a further aspect of the invention, a method is provided forcontrolling the output of a tunable acoustic gradient index ofrefraction optical lens. The method includes providing a tunableacoustic gradient index of refraction lens having a refractive indexthat varies in response to an applied electrical driving signal, andirradiating the optical input of the lens with a source ofelectromagnetic radiation. In addition the method includes driving thelens with a driving signal to control the index of refraction within thelens, and detecting the electromagnetic radiation output from the drivenlens with a detector. The method then includes providing a synchronizingsignal to the detector to select a time to detect the electromagneticradiation output from the driven lens when a desired refractive indexdistribution is present within the lens.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing summary and the following detailed description of thepreferred embodiments of the present invention will be best understoodwhen read in conjunction with the appended drawings, in which:

FIG. 1A schematically illustrates an exploded view of an exemplaryconfiguration of a TAG lens in accordance with the present inventionhaving a cylindrical shape;

FIG. 1B schematically illustrates an isometric view of assembled the TAGlens of FIG. 1A;

FIG. 2 schematically illustrates an exploded view of an exemplaryconfiguration of a TAG lens similar to that of FIG. 1A but having apiezoelectric tube that is segmented along the longitudinal axis;

FIGS. 3A and 3B schematically illustrate an exploded side-elevationalview and isometric view, respectively, of another exemplaryconfiguration of a TAG lens in accordance with the present invention;

FIG. 4A schematically illustrates an isometric view of another exemplaryconfiguration of a TAG lens in accordance with the present inventionhaving a rectangular shape;

FIG. 4B schematically illustrates the rectangular center casing of thelens of FIG. 4A;

FIG. 5 illustrates a characteristic pattern created by illuminating acircular TAG lens with a wide Gaussian collimated laser beam;

FIG. 6 illustrates the dependence of the peak refractive power of thelens, RP_(A), on the inner radius of the lens, r₀, assuming resonantdriving conditions;

FIG. 7 illustrates the dependence of the peak refractive power of thelens, RP_(A), on the static refractive index, n₀, assuming resonantdriving conditions;

FIG. 8 illustrates the dependence of the peak refractive power, RP_(A),on fluid sound speed, c_(s), assuming resonant driving conditions;

FIG. 9 illustrates the dependence of refractive power on drivingfrequency f=ω/(2π), assuming resonant driving conditions;

FIG. 10 illustrates the nonresonant dependence of the refractive powerof the lens, RP_(A), on the inner radius of the lens, r₀;

FIG. 11 illustrates the nonresonant dependence of refractive power,RP_(A), on fluid sound speed, c_(s);

FIG. 12 illustrates the nonresonant dependence of refractive power ondriving frequency f=ω/(2π);

FIG. 13 schematically illustrates a flow chart of a process for solvingthe inverse problem of specifying the driving waveform required toproduce a desired refractive index profile;

FIG. 14 schematically illustrates a ray diagram showing a TAG lensacting as a simple converging lens;

FIG. 15 illustrates both the goal and the actual deflection angle as afunction of radius at time t=0;

FIG. 16 illustrates both the goal and the actual refractive index as afunction of radius at time t=0;

FIG. 17 illustrates both the continuous and discretized spatialfrequencies of n_(goal)(r);

FIG. 18 illustrates one period of the time domain voltage signalrequired to generate the actual lensing effects portrayed in FIGS. 15and 16;

FIGS. 19A and 19B illustrate characteristic TAG-generated multiscaleBessel beams, with each figure showing two major rings plus the centralmajor spot, FIG. 19A showing the pattern at a low driving amplitude (30V) without minor rings and FIG. 19B showing the pattern at a higherdriving amplitude (65 V) with many minor rings;

FIGS. 20A and 20B schematically illustrate an experimental setup used tostudy the TAG beam characteristics and the coordinate system utilized,respectively;

FIG. 21A illustrates the predicted index profile at one instant in time,with a linear approximation to the central peak (dashed line);

FIG. 21B illustrates the predicted index profile one half-period laterin time than that shown in FIG. 21A, with linear approximations made tothe two central peaks (dashed line), with the scale of the spatial axisset by the driving frequency, in this case, 497.5 kHz;

FIGS. 21C and 21D illustrate theoretical predictions for theinstantaneous intensity patterns corresponding to a and b, respectively,observed with 355 nm laser light 50 cm behind the TAG lens withn_(A)=1.5×10⁻⁵ and scale bars set at 2 mm long;

FIGS. 21E and 21F show stroboscopic experimental images obtained inconditions identical to those of FIGS. 21C and 21D with the laserrepetition rate synchronized to the TAG driving frequency and TAG lensdriving amplitude of 5 V;

FIG. 22 illustrates the experimentally determined time-average intensityenhancement and propagation of the TAG central spot and first major ringwith the lens driven at 257.0 kHz with an amplitude of 37.2 V, and withthe x and z axes having significantly different scales—note thecharacteristic fringe patterns emanating from each peak in the indexprofile (cf. FIG. 21);

FIG. 23 illustrates the experimental and theoretical intensity profileof the TAG beam imaged 70 cm behind the lens showing that the fringepatterns extend similar to what one would expect from an axicon, withthe lens driven at 257.0 kHz with an amplitude of 37.2 V, and for thetheory, the value of n_(A) is 4×10⁻⁵;

FIG. 24 illustrates the beam divergence of the theoretical TAG,experimental TAG, Gaussian, and exact Bessel beams, with the TAG andGaussian beams achieve their maximum intensity approximately 58 cmbehind the lens, all beams having the same beam width at this location,and with the TAG lens driven at 257.0 kHz with an amplitude of 37.2 V,and for the theory, the value of n_(A) is 4×10⁻⁵;

FIG. 25 illustrates propagation similar to FIG. 22 with a 1.25 mmdiameter circular obstruction placed 27 cm behind the lens, with the TAGlens driven at 332.1 kHz with an amplitude of 5 V;

FIG. 26 illustrates experimental and theoretical locations of the firstmajor ring as a function of driving frequency, with the solid linerepresenting the theory given by Eq. 69, the squares representing thistheory, but also account for deflection in optical propagation due tothe asymmetry of the refractive index on either side of the major ring,and with the remaining symbols representing experimental results fromvarious trials;

FIG. 27A illustrates experimental variation in the intensity enhancement50 cm behind the lens as a function of driving amplitude, with the TAGlens driven at 257.0 kHz;

FIG. 27B illustrates theoretical variation in the intensity enhancement50 cm behind the lens as a function of driving amplitude when drivingthe lens at 257.0 kHz, showing good agreement with FIG. 27A;

FIG. 28 illustrates an experimental and theoretical central spot size asa function of driving amplitude when the lens is driven at 257.0 kHz andthe beam is imaged 50 cm behind the lens, with error bars representingthe size of a camera pixel;

FIG. 29 schematically illustrates an experimental setup with a pair oflenses L₁ and L₂ forming a telescope to reduce the size of the beam formicromachining, and with the delay between the AC signal and the laserpulse set by a pulse delay generator, the inset image on the leftshowing the spatial profile of the incident Gaussian beam, and the insetimage on the right showing the resulting annular beam after passingthrough the TAG lens;

FIG. 30 illustrates intensity images of instantaneous patterns obtainedby changing the TAG lens driving frequency, with pictures taken at adistance of 50 cm away from the lens, and frequencies from left to rightof 719 (bright spot), 980, 730, 457, 367, 337 kHz and amplitude of thedriving signal fixed at 9.8 V peak to peak; the ring diameter listed atthe bottom, and there is a half a period phase shift between the spotand the ring pattern;

FIGS. 31A and 31B illustrate a micromachined ring on the surface of apolyimide sample, with FIG. 31A showing an optical micrograph ofmicromachined ring structure and FIG. 31B showing the profilometryanalysis through the dashed line in FIG. 31A demonstrating that materialis removed over a depth of approximately 0.9 μm with little recastmaterial;

FIG. 32 illustrates a different intensity distribution at each laserspot, with a two pattern basis and demagnification of 50× used, and withthe lens is driven at 989 kHz in the upper image and driven at 531 kHzin the lower image;

FIGS. 33A and 33B illustrate the relation between driving voltage, andring radius, FIG. 33A, and number of rings, FIG. 33B;

FIG. 34 schematically illustrates the index of refraction in acylindrical TAG lens, which is a zeroth order Bessel function due to theacoustic wave in a cylindrical geometry and showing that as light entersthis modulated-index field, it will be bent according to the localgradients;

FIG. 35 schematically illustrates an experimental setup of a TAG systemin accordance with the present invention for dynamic focusing andimaging of an object of interest;

FIG. 36 illustrates images of the object of interest at three objectlocations as a function of TAG driving signal and laser pulse timing forthe TAG system of FIG. 35;

FIG. 37 schematically illustrates another experimental setup of a TAGsystem in accordance with the present invention for dynamic focusing andimaging of an object of interest;

FIGS. 38A-38C illustrate images of the object of interest at threeobject locations, respectively, as a function of TAG driving signal andlaser pulse timing for the TAG system of FIG. 37;

FIG. 39 illustrates images of a Bessel beam taken 70 cm behind the lensof FIGS. 3A, 3B to illustrate the time from which the driving frequencyis first changed to which the beam reaches a steady state, with thedriving frequency being 300 kHz and amplitude 60 V_(p-p), and each imageexposed for 0.5 ms;

FIG. 40 illustrates the intensity of a TAG lens generated beam withrespect to time after the driving voltage is switched off at t=0;

FIG. 41 illustrates three plots with differing viscosities of switchingspeed with respect to the driving frequency;

FIG. 42 illustrates the time-average output pattern from the lens ofFIGS. 4A, 4B with the periodicity of the spots on the order of 0.1 mm;

FIG. 43 illustrates a theoretical plot of an instantaneous pattern froma rectangular TAG lens driven at a frequency of 250 kHz, with theamplitude of the refractive index wave (both horizontal and vertical)being 3.65×10⁻⁵; and

FIG. 44 illustrates the time-averaged pattern of the theoretical plot ofFIG. 31.

DETAILED DESCRIPTION OF THE INVENTION

Referring now to the figures, wherein like elements are numbered alikethroughout, FIGS. 1B, 1B schematically illustrates an exemplaryconfiguration of a TAG lens 100 in accordance with the present inventionhaving a cylindrical shape. The TAG lens 100 is a piezoelectricallydriven device that uses sound waves to modulate the wavefront of anincident light beam. The lens 100 is composed of a hollow piezoelectrictube 10 that is constrained by two transparent windows 30 on either endfor optical access and filled with a refractive material, such as a gas,solid, liquid, plasma, or optical gain medium, for example. The TAG lens100 works by creating a standing acoustic wave in the refractive liquid.The acoustic standing wave is created by applying an alternatingvoltage, typically in the radio-frequency range, to the piezoelectrictube 10 by a controller 90. The controller 90 may include a functiongenerator passed through an RF amplifier and impedance matching circuit.

Turning to FIGS. 1A, 1B in more detail, the TAG lens 100 includes apiezoelectric tube 10 having a generally cylindrical shape, though othershapes such as a square, triangular, hexagonal cross-section, etc. maybe used by utilizing multiple piezoelectric elements as described below.The piezoelectric tube 10 includes an outer electrode 14, and an innerelectrode which may be wrapped from the inside surface of thepiezoelectric tube 10 to the outer surface of the tube 10 (using aconductive copper tape adhered to the inside and wrapped around to theoutside, for example) to provide an annular electrode contact region 12for the inner electrode. The annular electrode contact region 12 and theouter electrode 14 may be electrically separated from one another by anannular gap 15 disposed therebetween. While a single piezoelectric tube10 is shown, multiple tubes can be used end-to-end or a singlepiezoelectric tube 10 a can be segmented along the longitudinal axisinto different zones 14 a, 14 b which can be separately electricallyaddressed and driven to permit separate electrical signals to bedelivered to each tube or zone 14 a, 14 b, FIG. 2. For example, thepiezoelectric tube 10 a may include an annular gap 15 b disposed betweenthe two ends of the tube 10 a to electrically isolate two outerelectrode zones 14 a, 14 b from one another along the axis of the tube10 a. A similar electrical gap may be provided internally to the tube 10a to electric isolate to inner electrode zones. Each of the separatedinner electrode zones may be wrapped to a respective end of the tube 10a to provide a respective annular electrode contact region 12 a, 12 bdisposed at opposing ends of the tube 10 a. Each of the separatelongitudinal zones 14 a, 14 b may then be driven by a separate signal.Such a configuration can, for example, permit a single TAG lens to beoperated as if it were a compound lens system, with each tube orlongitudinal zone 14 a, 14 b corresponding to a separate optical elementor lens. Additionally, the piezoelectric tube 10 may be segmentedcircumferentially so that multiple electrically addressable zones mayexist at a given longitudinal location. Likewise, electricallyaddressable zones that have an arbitrary shape and size may be providedon the inner and outer cylindrical surfaces of the piezoelectric tube10, where a given zone on the inner cylindrical surface may (or may not)coincide with an identical zone on the outer surface.

A cylindrical gasket 18 having an inner diameter larger than the outerdiameter of the piezoelectric tube 10 may be provided to slide over thepiezoelectric tube 10 to center and cushion the piezoelectric tube 10within the rest of the structure. An opening, such as slot 19, may beprovided in the cylindrical spacer gasket 18 to permit access to thepiezoelectric tube 10 for purpose of making electrical contact with thepiezoelectric tube 10 and filling the interior of the piezoelectric tube10 with a suitable material, e.g., a fluid (liquid or gas). The spacergasket 18 may be housed within a generally cylindrical inner casing 20which may include one or more fluid ports 22 in the sidewall throughwhich fluid may be introduced into or removed from the inner casing 20and the interior of the piezoelectric tube 10 disposed therein. One ormore outlet/inlet ports 44 having barbed protrusions may be provided inthe fluid ports 22 to permit tubing to be connected to the outlet/inletports 44 to facilitate the introduction or removal of fluid from the TAGlens 100. In this regard the inner casing 20, spacer gasket 18, andpiezoelectric tube 10 are configured so that fluid introduced throughthe inlet port 44 may travel past the spacer gasket 18 and into theinterior of the piezoelectric tube 10. In addition, one or moreelectrical feedthrough ports 24 may be provided in a sidewall of theinner casing 20 to permit electrical contact to be made with thepiezoelectric tube 10. For instance, wires may be extended through theelectrical feedthrough ports 24 to allow electrical connection to thepiezoelectric tube 10.

At either end of the inner casing 20 transparent windows 30 may beprovided and sealed into place to provide a sealed enclosure forretaining a refractive fluid introduced through the inlet port 44 withinthe inner casing 20. To assist in creating a seal, an O-ring 26 mayprovided between the ends of the inner casing 20 and the transparentwindows 30, and the end of the inner casing 20 may include an annulargroove into which the O-rings 26 may seat. Likewise spacer O-rings 16may be provided between the ends of the piezoelectric tube 10 and thetransparent windows 30. The windows 30 may comprise glass or any otheroptical material that is sufficiently transparent to the electromagneticwavelengths at which the lens 100 is to be used. For instance, thewindows 30 may be partially mirrored, such to be 50% transparent, forexample. In addition, the windows 30 may comprise flat slabs or mayinclude curved surfaces so that the windows 30 function as a lens. Forexample, one or both of the surfaces of either of the windows 30 mayhave a concave or convex shape or other configuration, such as a Fresnelsurface, to introduce optical power. Further, the windows 30 may beconfigured to manipulate the incident optical radiation in othermanners, such as filtering or diffracting.

The inner casing 20 and transparent windows 30 may be dimensioned to fitwithin an outer casing 40 which may conveniently be provided in the formof a 2 inch optical tube which is a standard dimension that can bereadily mounted to existing optical components. To secure the innercasing 20 within the outer casing 40, the outer casing 40 may include aninternal shoulder against which one end of the inner casing 20 seats. Inaddition, the outer casing 40 may be internally threaded at the endopposite to the shoulder end. A retaining ring 50 may provided thatscrews into the outer casing 40 to abut against the end of the innercasing 20 to secure the inner casing 20 with and the outer casing 40.The outer casing 40 may include an access port 42, which may be providedin the form of a slot, through which the inlet/outlet ports 44 andelectrical connections, such as a BNC connector 46, may pass. In orderto supply the driving voltage to lens the 100, a controller 90 may beprovided in electrical communication with the connector 46, which inturn is electrically connected to the piezoelectric tube 10, via theannular electrode contact region 12 and outer electrode 14, for example.

Turning next to FIGS. 3A and 3B, an additional exemplary configurationof a cylindrical TAG lens 300 in accordance with the present inventionis illustrated. Among the differences of note between the TAG lens 300of FIGS. 3A and 3B in the TAG lens 100 of FIGS. 1A and 1B are the mannerin which electrical contact is made with the inner surface of thepiezoelectric tube 310 and the relatively fewer number of parts. The TAGlens 300 includes a piezoelectric tube 310 which may be similar inconfiguration to the piezoelectric tube 10 of the TAG lens 100. In orderto make contact with the inner electrode surface 312, an inner electrodecontact ring 320 may be provided that includes an inner electrodecontact tab 322 which may extend into the cavity of the piezoelectrictube 310 to make electrical contact with the inner electrode surface312. To prevent electrical communication between the inner electrodecontact ring 320 and the outer electrode 314, an annular insulatinggasket 352 may be provided between the piezoelectric tube 310 and theinner electrode contact ring 320.

To create a sealed enclosure internal to the piezoelectric tube 310 inwhich a refractive fluid may be contained, two housing end plates 340may be provided to be sealed over the ends of the piezoelectric tube310. In this regard, annular sealing gaskets 350 may be provided betweenthe ends of the piezoelectric tube 310 and the housing end plates 340 tohelp promote a fluid-tight seal. The housing end plates 340 may includea cylindrical opening 332 through which electromagnetic radiation maypass. In addition, the housing end plates 340 may include windows 330disposed within the opening 332, which may include a shoulder againstwhich the windows 330 seat. A refractive fluid may be introduced andwithdrawn from the lens 300 through optional fill ports 344, or byinjecting the refractive fluid between the sealing gasket 350 and thehousing end plates 340 using a needle. An electrical driving signal maybe provided by a controller 390 which is electrically connected to theouter electrode 314 and the inner electrode contact ring 320 by wires316 to drive the piezoelectric tube 310. Like the controller 90 of FIG.1A, the controller 390 may include a function generator passed throughan RF amplifier and impedance matching circuit. Though the lens 300 ofFIG. 3 contains fewer parts than the lens 100 of FIG. 1A, the lens 100may be more convenient to use due to the increased ease with which thelens 100 may be filled and sealed. In addition, the electrodeconfiguration of the lens 100, specifically the inclusion of the annularelectrode contact region 12 for making electrical contact with the innerelectrode of the piezoelectric tube 10, may lead to the creation of moreaxisymmetric acoustic waves (i.e., about the longitudinal axis of thepiezoelectric tube 10) than would be possible with the point contactprovided by contact tab 322 of the lens 300.

Turning next to FIGS. 4A and 4B, an alternative exemplary configurationof a rectangular TAG lens 200 in accordance with the present inventionis illustrated. The lens 200 may include two piezoelectric plates 210oriented 90°. with respect to one another to provide two sides of thesquare cross-section of the rectangular enclosure, FIG. 3B. To completethe square cross-sectional shape of the lens cavity two planar walls 212may be provided opposite the two piezoelectric plates 210. Providing twopiezoelectric plates 210 can be useful for generating arbitrary patternsby combining several input signals that generate two independentorthogonal wavefronts, and as such is not limited to circularlysymmetric patterns. Electrical wires 216, 218 may be connected toopposing sides of the piezoelectric plates 210, with the “hot” wires 216electrically connected to the surface of the piezoelectric plate 210closest to the interior of the lens 200, FIG. 3B. The wires 216, 218 mayin turn be electrically connected with a controller that provides thedriving voltage for the piezoelectric plates 210.

The piezoelectric plates 210 and planar walls 212 are enclosed within acenter casing 240 which may have threaded holes through which adjustmentscrews 214 may pass to permit adjustment of the location of the walls212. The piezoelectric plates 210 in turn may be secured with anadhesive 213 to the center casing 240 to secure them in place. Sealingwashers 215 may be provided internally to the center casing 240 on theadjustment screws 214 to help seal a refractive fluid, F, within thecenter casing 240. The center casing 240 may be provided in the form ofan open-ended rectangular tube, to which two end plates 220 may beattached to provide a sealed enclosure 250 in which the refractivefluid, F, may be retained. Attachment may be effected through the meansof bolts 242, or other suitable means. The bolts 242 pass through theend plates 220 and center casing 240. To aid in providing a fluid-tightseal between the center casing 240 and the end plates 220, sealinggaskets 226 may be provided between each end face of the center casing240 and the adjoining end plate 220. The electrical wires 216, 218 maypass between the sealing gaskets 226 and the center casing 240 or endplates 220. The end plates 220 may also include a central square opening232 in which transparent windows 230 may be mounted (e.g., with anfluid-tight adhesive or other suitable method) to permit opticalradiation to pass through the lens 200 and the refractive fluid, F, inthe central enclosure 250. The refractive fluid, F, may be introducedinto the sealed enclosure 250 via fluid ports or by injecting therefractive fluid, F, into the sealed enclosure 250 by inserting a needlebetween the sealing gasket 226 and the center casing 240 or end plate220. The particular exemplary lens 200 fabricated in tested (results inFIG. 42) was 0.5 inches thick, and 2 by 2 inches in the other twodimensions, and used PZT5A3, poled with silver electrodes, MorganElectro Ceramics as the piezoelectric plates 210. Each of thepiezoelectric plates 210 were driven at the same frequency andamplitude, and were in phase. Silicone oils of 0.65 and 5 cS have beenused successfully. The patterns seen in FIG. 42 may be seen over a rangefrequencies (200-1000 kHz) and driving amplitudes (5-100 V_(p-p)), e.g.400 kHz and 30 V_(p-p).

Having provided various exemplary configurations of TAG lenses 100, 200,300 in accordance with the present invention, discussion of theiroperation follows.

I. Operation of Tag Lens

A predictive model for the steady-state fluid mechanics behind TAGlenses 100, 200, 300 driven with a sinusoidal voltage signal ispresented in this section. The model covers inviscid and viscous regimesin both the resonant and off-resonant cases. The density fluctuationsfrom the fluidic model are related to refractive index fluctuations. Theentire model is then analyzed to determine the optimal values of lensdesign parameters for greatest lens refractive power. These designparameters include lens length, radius, static refractive index, fluidviscosity, sound speed, and driving frequency and amplitude. It is foundthat long lenses 100, 200, 300 filled with a fluid of high refractiveindex and driven with large amplitude signals form the most effectivelenses 100, 200, 300. When dealing with resonant driving conditions, lowdriving frequencies, smaller lens radii, and fluids with larger soundspeeds are optimal. At nonresonant driving conditions, the opposite istrue: high driving frequencies, larger radius lenses, and fluids withlow sound speeds are beneficial. The ease of tunability of the TAG lens100, 200, 300 through modifying the driving signal is discussed, as arelimitations of the model including cavitation and nonlinearities withinthe lens 100, 200, 300.

The TAG lens 100, 200, 300 uses acoustic waves to modulate the densityof an optically transparent fluid, thereby producing a spatially andtemporally varying index of refraction—effectively a time-varyinggradient index lens 100, 200, 300. Because the TAG lens 100, 200, 300operates at frequencies in the order of 10⁵ Hz, the patterns observed(FIG. 5) by passing a CW collimated laser beam through the TAG lens 100,200, 300 are time-average images of a temporally periodic pattern. Theminor rings around each bright major ring approximate nondiffractingaxicon-generated Bessel beams. The mechanics behind these patterns isexplained below, and the optic of pattern formation are discussed insection III below. The square TAG lens 200 has been seen to producespatterns such as those in FIG. 42.

An exemplary TAG lens 300 used in the analyses of this section isillustrated in FIGS. 3A, 3B, and may comprise a cylindrical cavityformed by a hollow piezoelectric tube 310 with two flat transparentwindows 330 on either side for optical access. The cavity may be filledwith a refractive fluid and the piezoelectric tube 310 driven with an ACsignal generating vibration in several directions, the important ofwhich is the radial direction. This establishes standing-wave densityand refractive index oscillations within the fluid, which are used toshape an incident laser beam.

Based on the TAG lens 300 configuration of FIGS. 3A, 3B, a predictivemodel is developed for the fluid mechanics and local refractive indexthroughout the lens 300 under steady-state operation. This model is alsoexpected to generally apply to the cylindrical lens 100 of FIGS. 1A, 1B,2. (A previous model for an acoustically-driven lens had been proposed,however this model had invoked time-invariant nonlinear acoustictheories which ignore the more significant linear effects occurring inthese lenses. Cf. Higginson, et al., Applied Physics Letters, 843(2004).) Experimentation has shown that TAG beams are stronglytime-varying, and the following linear acoustic model better explainsall characteristics of the TAG lens 300. The results of the presentmodel will be provided using “base case” TAG lens parameters. Optimizingthe refractive capabilities of the lens 300 relative to this base casefor desired applications is also discussed. The effects of modifying thelens dimensions, filling fluid, and driving signal are all examined, aswell as how modifying the driving signal can be used to tune the indexof refraction within the lens 300.

Base Case Parameters

FIG. 5 shows the pattern generated by a “base case” TAG lens 300 (exceptfor a driving frequency shifted to 299.7 kHz) as observed 80 cm behindthe lens 300. The lens 300 itself is diagrammed in FIGS. 3A, 3B. Thebase case parameters for this lens 300 are listed in Table I.

TABLE I Base case parameters for the TAG lens, divided into geometric,fluid, and driving signal parameters, respectively. Parameter SymbolBase Case Value Lens inner radius r_(s) 3.5 cm Lens length L 4.06 cmFluid Viscosity V 100 cS Static refractive Index n_(s) 1.4030 Speed ofsound

1.00

Fluid Density p

964 kg/m

Voltage Amplitude V_(A) 10 V Peak inner wall velocity v_(A) 1 cm/sResonant Frequency f 246.397 kHz Off-Resonant Frequency f 253.5 kHz

indicates data missing or illegible when filed

The piezoelectric material used for the tube 310 is lead zirconatetitanate, PZT-8, and the filling fluid for the lens 300 is a Dow Corning200 Fluid, a silicone oil. The piezoelectric tube 310 is driven by thecontroller 390 which includes a function generator (Stanford ResearchSystems, DS345) passed through an RF amplifier (T&C Power Conversion, AG1006) and impedance matching circuit, which can produce AC voltages upto 300 V_(pp) at frequencies between 100 kHz and 500 kHz. Otherimpedance matching circuits could be used to facilitate differentfrequency ranges. Two different driving frequencies are used,corresponding to resonant and off-resonant cases, listed in Table I.

Mechanics Piezoelectric Transduction

As indicated above, the piezoelectric transducer used to drive the TAGlens 300 comes in the form of a hollow cylinder or tube 310. Theelectrodes 312, 314 are placed on the inner and outer circumferences ofthe tube 310. The driving voltage frequency and amplitude is applied tothe piezoelectric tube 310 so that

$\begin{matrix}{\mspace{79mu} {V = {V\text{?}{{\sin ( \text{?} )}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (1)\end{matrix}$

The theory behind how a hollow piezoelectric tube 310 will respond tosuch a driving voltage has already been published (Adelman, et al.,Journal of Sound and Vibration, 245 (1975)), which leads to inner wallvelocities on the order of V_(A)=1 cm/s, assuming driving voltageamplitudes on the order of 10 V. It is important to note that the wallvelocity is always proportional to the driving voltage amplitude.

Fluid Mechanics

The mechanics of the fluid within the lens 300 is described by threeequations: conservation of mass, conservation of momentum, and anacoustic equation of state. Stated symbolically, these equations are:

$\begin{matrix}{\mspace{79mu} {{{\frac{\partial\rho}{\partial} + {\nabla{\cdot ( {\rho \; v} )}}} = 0},}} & (2) \\{\mspace{79mu} {{{{\frac{\partial}{\partial t}( {\rho \; v} )} + {\nabla\rho} + {\nabla{\cdot ( {\rho \; {v \otimes v}} )}} + {V \cdot D}} = 0},}} & (3) \\{\mspace{79mu} {{p = {p_{0} = {\text{?}( {p - p_{0}} )}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (4)\end{matrix}$

where {circle around (x)} represents the tensor product and D is theviscous stress tensor whose elements are given by

$\begin{matrix}{\mspace{79mu} {\text{?} = {{{- ( {\eta - {2{\mu/3}}} )}( {\nabla{\cdot v}} )\text{?}} - {{{\mu ( {\frac{\partial\text{?}}{\partial\text{?}} + \frac{\partial\text{?}}{\partial\text{?}}} )}.\text{?}}\text{indicates text missing or illegible when filed}}}}} & (6)\end{matrix}$

Here, ρ is the local density, v is the local fluid velocity, p is thelocal pressure, μ is the dynamic shear viscosity, and η is the dynamicbulk viscosity. Bulk viscosities are not generally tabulated and aredifficult to measure. For most fluids, η is the same order of magnitudeas μ. For the base case, it is assumed that η=μ. Equation 4 assumessmall amplitude waves where c_(s) is the speed of sound within the fluidat the quiescent density and pressure, ρ₀ and ρ₀. This equationrepresents the linearized form of all fluid equations of state.

Substituting the equation of state (Eq. 4) into the momentumconservation equation (Eq. 3) yields two coupled differential equationsfor the dependent variables ρ and v. Applying no-slip conditions at theboundaries of the cell translates to these boundary conditions:

$\begin{matrix}{\mspace{79mu} {{{v\text{?}} = {v\text{?}{\cos ( \text{?} )}\text{?}}},}} & (6) \\{\mspace{79mu} {{{v\text{?}} = {{v\text{?}} = 0.}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (7)\end{matrix}$

The radial boundary condition is determined from the velocity of theinner wall 312 of the piezoelectric tube 310. This assumes that thepiezoelectric tube 310 is stiff compared to the fluid and that acousticwaves within the fluid do not couple back into the piezoelectric motion.Impedance spectroscopy conducted on the TAG lens 300 shows that exceptnear resonances, the TAG lens 300 impedance is the same regardless ofthe filling fluid chosen. Hence, this assumption is generally true,however some corrections may be needed when near resonance. The presenceof the dead space-created by the sealing gaskets 350 between thepiezoelectric tube ends and windows 330, especially in the configurationof FIGS. 3A, 3B, will also modify the boundary condition in Eq. 6,however this effect is neglected, because it is expected to only besignificant near the sealing gaskets 350 themselves.

Typically, a unique solution for ρ and v at all times would require twoinitial conditions as well as the above boundary conditions. However forthe steady-state response to the vibrating wall, the initial conditionsdo not affect the steady-state response.

The following assumptions reduce the dimensionality of the problem,making it more tractable. First, the azimuthal dependence can beeliminated because of the lack of angular dependence within the boundaryconditions (Eqs. 6 and 7). Second, the z-dependence of the boundaryconditions only appears in the no-slip conditions at the transparentwindows 330. Physically, this effect is expected to be localized to aboundary layer of approximate thickness

$\begin{matrix}{\delta = {\sqrt{\frac{2\mu}{\rho_{0}\omega}}.}} & (8)\end{matrix}$

For the base case parameters, this thickness comes out to approximately10 μm. Thus, the lens 300 is operating in the limit δ<<L, and solvingthe problem outside the boundary layer will account for virtually allthe fluid within the lens 300. Furthermore, because radial gradients areexpected to be reduced within the boundary layer, the boundary layereffect can be approximated by simply using a reduced effective lenslength. Gradients in the z-direction are expected to be much largerwithin the boundary layer because the fluid velocity transitions to zeroat the wall. However for a normally incident beam of light, all that issignificant is the transverse gradient in total optical path lengththrough the lens 300. Optical path length differences due to densitygradients in the z-direction within the thin boundary layer areinsignificant compared to the optical path length differences within thebulk. The result of these considerations is that an approximate solutioncan be found by solving the one dimensional problem, assuming ρ is onlya function of r and then applying that solution to all values of zwithin the lens 300.

The problem can be further simplified by linearization. This assumesthat the acoustic waves have a small amplitude relative to staticconditions. Each variable is expanded in terms of an arbitrary amplitudeparameter, λ:

$\begin{matrix}{\mspace{79mu} {{\rho ( {r,t} )} = {{\rho \text{?}} + {\lambda \; \rho \text{?}( {r,\text{?}} )} + {\lambda \text{?}\rho \text{?}( {r,t} )} + \ldots}}} & (9) \\{\mspace{79mu} {{{\text{?}( {r,t} )} = {0 + {\lambda \; v\text{?}( {r,t} )} + {\lambda \text{?}v\text{?}( {r,t} )} + \ldots}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (10)\end{matrix}$

Furthermore, the wave amplitudes are assumed small and therefore anysecond order or higher term (λ², λ³, etc.) is much less than the zerothor first order terms, so the higher order terms can be dropped from theequations. Keeping only the zeroth and first order terms results in ρ(r,t)=ρ₀+λp₁(r, t) and v(r, t)=λv₁(r, t), and Eqs. 2 and 3 can be rewrittenas:

$\begin{matrix}{\mspace{79mu} {{\lambda \lbrack {\frac{\partial\text{?}}{\partial t} + {\text{?} \cdot ( \text{?} )}} \rbrack} = 0.}} & (11) \\{\mspace{79mu} {{{\lambda \lbrack {{\frac{\partial}{\partial t}( \text{?} )} + {\text{?}\text{?}\text{?}} + {\text{?} \cdot \text{?}}} \rbrack} = 0},{\text{?}\text{indicates text missing or illegible when filed}}}} & (12)\end{matrix}$

where D₁ is defined in the same way as D in Eq. 5, except with vreplaced by v₁.

Inviscid Solution

One solution of interest is the inviscid solution because it reasonablyaccurately predicts the lens output patterns for low viscosities inoff-resonant conditions while retaining a simple analytic form. Thissolution is found by setting μ=η=0. In the one dimensional case, theproblem becomes:

$\begin{matrix}{\mspace{79mu} {{{\frac{\text{?}}{\partial t} + {\frac{1}{r}\frac{\partial}{\partial r}( {r\text{?}v} )}} = 0},}} & (13) \\{\mspace{79mu} {{{{\frac{\partial}{\partial t}( {\rho_{o}v} )} + {\text{?}\frac{\partial\rho_{1}}{\partial r}}} = 0},}} & (14) \\{\text{?} = {\text{?}{{\cos ( \text{?} )}.\text{?}}\text{indicates text missing or illegible when filed}}} & (15)\end{matrix}$

It can be directly verified by substitution that the solution to thisproblem is

$\begin{matrix}{{{\text{?}( {r,t} )} = {\text{?}( \text{?} ){\sin ( {\omega \text{?}} )}}},} & (16) \\{\mspace{79mu} {{{v( {r,\text{?}} )} = {{- \text{?}}( {\omega \; {r/\text{?}}} ){\cos ( {\omega \text{?}} )}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (17)\end{matrix}$

where ρ_(A)=−(ρ_(o)ν_(A))/(c_(s)J₁(ωr₀/c_(s))). For the base caseoff-resonant frequency, ρ_(A) is expected to be 0.090 kg/m³.

Viscous Solution

An effective kinematic viscosity is defined as ν′=(η+4μ/3). In caseswhere this viscosity is large compared to c_(s) ²/ω or when the lens 300is driven near a resonant frequency of the cavity, viscosity becomessignificant and the solution is somewhat more complex. To put theviscosity threshold in context, the base case fluid, 100 cS siliconeoil, is considered low viscosity for frequencies f<<c_(s) ²/(2πν′)=700MHz.

Differentiating Eq. 11 with respect to time and taking the divergence ofEq. 12, the equations can be decoupled and all dependence on veliminated to yield the damped wave equation,

$\begin{matrix}{{{{\text{?}( {\text{?} + {v^{\prime}\frac{\partial p_{i}}{\partial t}}} )} - \frac{\partial^{2}p_{1}}{\partial t^{2}}} = 0.}{\text{?}\text{indicates text missing or illegible when filed}}} & (18)\end{matrix}$

By evaluating Eqs. 11 and 12 at r=r₀ and assuming a curl-free velocityfield there, Eq. 6 can be converted from a boundary condition invelocity to the following Neumann boundary condition in density,

$\begin{matrix}{\mspace{79mu} {{\frac{\partial p_{i}}{\partial r}\text{?}} = {{\text{?}{\sin ( {\omega \; t} )}} - {\text{?}{{\cos ( {\omega \; t} )}.\text{?}}\text{indicates text missing or illegible when filed}}}}} & (19)\end{matrix}$

The steady-state one-dimensional solution to the above wave equation andboundary condition can be expanded as a sum of eigenfunctions:

$\begin{matrix}{{{\rho_{1}( {r,t} )} = {{r( {{A\; {\sin ( {\omega \; t} )}} + {B\; {\cos ( {\omega \; t} )}}} )} + {\text{?}{{J_{0}( {\text{?}r} )}\lbrack {{\text{?}{\sin ( {\omega \; t} )}} + {D_{m}{\cos ( {\omega \; t} )}}} \rbrack}}}},{\text{?}\text{indicates text missing or illegible when filed}}} & (20)\end{matrix}$

where k_(m)=x_(m)/r₀ with x_(m) being the location of the m^(th) zero ofJ₁(x) and x₀=0. A and B can be found by substituting this solution intoEq. 19. C_(m) and D_(m) can be found by substituting the solution intoEq. 18 and integrating against the orthogonal eigenfunction J₀(k_(n)r)over the entire circular domain. The resulting expressions are:

$\begin{matrix}{\mspace{79mu} {A = \frac{\rho_{0}v_{A}\omega \; \text{?}}{{v^{\prime 2}\omega^{2}} + \text{?}}}} & (21) \\{\mspace{79mu} {{B = \frac{\rho_{0}v_{A}\omega^{2}v^{\prime}}{{v^{\prime 2}\omega^{2}} + \text{?}}},}} & (22) \\{{\text{?} = ( \text{?} )}{\text{?},}} & (23) \\{\mspace{79mu} {{D_{m} = ( \text{?} )}\mspace{79mu} {\omega \; v^{\prime}\text{?}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (24)\end{matrix}$

In the expressions above, E_(m) and F_(m) are the nondimensionalintegrals,

$\begin{matrix}{\mspace{79mu} {{{E\text{?}} = {\int_{0}^{\text{?}}{\text{?}J\text{?}( {x\text{?}x} )\ {x}}}},}} & (25) \\{\mspace{79mu} {{V\text{?}} = {\int_{0}^{\text{?}}{{J_{0}( {x\text{?}x} )}\ {{x}.\text{?}}\text{indicates text missing or illegible when filed}}}}} & (26)\end{matrix}$

By taking the limit ν′→0 and using the same trick of integrating againstan orthogonal eigenfunction, the inviscid solution in Eq. 16 can berecovered.

Resonant Driving Conditions

Another important limit is that of operating near a resonance of thecavity using a relatively low viscosity fluid. Operating at the n^(th)(>0) resonance means that ω=c_(s)k_(n). Note that at resonantfrequencies, the inviscid solution in Eq. 16 diverges becauseJ₁(kr_(o))→0 in the denominator of ρ_(A). Consequently, in order to geta valid solution near resonance, the full viscous solution isnecessary—even at low viscosities. As discussed in the previous section,low viscosity means that ν′<<c_(s) ²/ω. In this limit, the coefficientsof the viscous solution look as follows:

$\begin{matrix}{\mspace{79mu} { Aarrow\text{?} ,}} & (27) \\{\mspace{79mu} { Barrow{{- v^{\prime}}\text{?}} ,}} & (28) \\{ \text{?}arrow\text{?} ,} & (29) \\{ \text{?}arrow{- \text{?}} ,} & (30) \\{ \text{?}arrow{{- v^{\prime}}\text{?}} {\text{?},}} & (31) \\ \text{?}arrow{{- \frac{1}{v^{\prime}}}{\text{?}.\text{?}}\text{indicates text missing or illegible when filed}}  & (32)\end{matrix}$

Note that as the viscosity vanishes, the only term that diverges is theD_(m=n) term. All the other terms either vanish or do not change. Thismeans that when driving on resonance with a low viscosity fluid only theD_(m=n), term is significant, and the solution for the density becomes,

$\begin{matrix}{ {\text{?}( {r,t} )}arrow{{- \frac{1}{v^{\prime}}}( \text{?} )} \mspace{20mu} {{J_{0}( \text{?} )}{{\cos ( {\omega \; t} )}.\text{?}}\text{indicates text missing or illegible when filed}}} & (33)\end{matrix}$

At the resonant base case frequency, the amplitude of ρ₁ takes the value9.1 kg/m³.

From Density to Refractive Index

The Lorentz-Lorenz equation can be used to determine the local index ofrefraction from the fluid density. This relationship is

$\begin{matrix}{{n = \sqrt{\frac{{2\; Q\; \rho} + 1}{1 - {Q\; \rho}}}},} & (34)\end{matrix}$

where Q is the molar refractivity, which can be determined from n_(o)and ρ_(o). For small ρ₁, this equation can be linearized by a Taylorexpansion about the static density and refractive index. Substitutingfor Q, this takes the form,

$\begin{matrix}{n = {n_{o} + {\frac{n_{o}^{4} + n_{o}^{2} - 2}{6\; n_{o}}{( \frac{\rho_{1}}{\rho_{0}} ).}}}} & (35)\end{matrix}$

In the resonant base case, the amplitude of oscillation of the densitystanding wave is less than 1% of the static density. Comparing the trueLorentz-Lorenz equation with the linearized version, one finds that theerror in refractive index due to linearization is less than 0.2%.

In the inviscid linearized acoustic case, the refractive index given byEq. 35 assuming the density distribution in Eq. 16 or 33, depending onresonance, reduces to an expression of the form,

$\begin{matrix}{\mspace{79mu} {{n = {{n\text{?}} + {n\text{?}J\text{?}({kr}){\sin ( \text{?} )}}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (36)\end{matrix}$in-the off-resonant case, or

$\begin{matrix}{\mspace{79mu} {{n = {{n\text{?}} + {n\text{?}J\text{?}({kr}){\cos ( \text{?} )}}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (37)\end{matrix}$

at resonance. The full expression for n_(A) in the low-viscosityoff-resonant case is:

$\begin{matrix}{\mspace{79mu} {{n_{A} = ( \text{?} )},{\text{?}\text{indicates text missing or illegible when filed}}}} & (38)\end{matrix}$

and in the resonant case n_(A) is given by:

$\begin{matrix}{\mspace{79mu} {{n_{A} = ( \text{?} )}{\text{?}\text{indicates text missing or illegible when filed}}}} & (39)\end{matrix}$

For the base case, n_(A) is expected to have an off-resonant value of4.3×10⁻⁵. On resonance, it is expected to have a base case value of4.3×10⁻³. Similar solutions can be obtained for the viscous case.

Optimizing the Figure of Merit: Refractive Power

In order to get the most out of a TAG lens 300 under steady stateoperation, one wishes to maximize the peak refractive power. The lowerbound is always zero, given by the static lens 300 without any inputdriving signal. Higher refractive powers increase the range ofachievable working distances and Bessel beam ring spacings. Therefractive power, RP, is defined here to be the magnitude of thetransverse gradient in optical path length. This is given by the productof the transverse gradient in refractive index and the length of thelens 300. Under thin lens and small angle approximations, the maximumangle that an incoming collimated ray can be diverted by the TAG lens300 is equal to its refractive power. For a simple converging lens, itsRP is also equal to its numerical aperture.

Maximizing the refractive power can be accomplished by altering thedimensions of the lens 300, the filling fluid, or the driving signal.Because the base case TAG lens 300 is well within the low viscosityrange of the parameter space, discussion in this section will be limitedto only low-viscosity fluids in the resonant and off-resonant cases sothat Eq. 36 or 37 applies with n_(A) given by Eq. 38 or 39.

The first step is to calculate the TAG lens peak refractive power,RP_(A), using Eq. 36 and assuming azimuthal symmetry within the lens300:

$\begin{matrix}{\begin{matrix}{\mspace{79mu} {{RP}_{A} = {\text{?}{{{\nabla{OPL}} \cdot \hat{r}}}}}} \\{= {\text{?}{{L{{\nabla n} \cdot \hat{r}}}}}} \\{= {\text{?}{{{{Lkn}_{A}\text{?}({kr})}}.}}}\end{matrix}{\text{?}\text{indicates text missing or illegible when filed}}} & (40)\end{matrix}$

Therefore, RP_(A) is maximized by maximizing |Lkn_(A)|, while the termJ₁(kr) only determines at what radial location this maximum is achieved.In order to maximize |Lkn_(A)|, each of the parameters in Eqs. 38 and 39is considered. Because the dependence on these parameters can varybetween resonant and off-resonant driving conditions, the analysis hasbeen divided into the two subsections below.

Optimizing Resonant Conditions

It is first assumed that the lens 300 will be driven under resonantconditions. This will yield the highest refractive, powers. At theresonant base case frequency, RP_(A) takes the value 0.16. The model forthis section uses the refractive index given by Eq. 37 with n_(A) givenby Eq. 39.

Optimizing Lens Dimensions

The size of the TAG lens 300 is considered first. This is determined bythe piezoelectric tube length L and inner radius r₀. The refractivepower of the TAG lens 300 is proportional to L, so longer lenses aredesirable. With increasing length, thin lens approximations will becomeincreasingly erroneous, and eventually the TAG lens 300 will function asa waveguide.

The dependence on transverse lens size is not a simple relationshipbecause of the Bessel functions in the denominator of Eq. 39 and thefact that the value of n in E_(n) and F_(n) depends on r₀. Therelationship between the refractive power and the inner lens radius isplotted in FIG. 6. This figure shows that on resonance, higherrefractive powers can be achieved with lenses having a smaller radius.Discrete points are plotted because resonance is only achieved atdiscrete inner radii. This effect can be attributed to increased viscouslosses due to increased acoustic wave propagation distance.

Optimizing the Refractive Fluid

The relevant properties of the refractive fluid include its static indexof refraction n₀, its effective kinematic viscosity V′, and the speed ofsound within the material, c_(s).

Increasing the value of n₀ affects only the first term of Eq. 39 andincreases the TAG lens refractive power. Due to the nature of theLorentz-Lorenz equation, the same fractional variation in density willhave a greater effect on the refractive index of a material with anaturally high refractive index than it will on a material with a lowerrefractive index. This effect is plotted in FIG. 7. It is clear thathigher static indices of refraction improve lens performance.

In the resonant case, the viscosity of the fluid is significant andlower viscosities are more desirable because the refractive indexamplitude is inversely proportional to the effective kinematicviscosity. In symbols, n_(A)∝ν′¹. This result is expected because lowerviscosities will mean less viscous loss of energy within the lens 300.

As with the inner radius, the effect of the sound speed on therefractive power cannot be easily analytically represented because ofthe Bessel functions in the denominator of Eq. 39 and the dependence ofE_(n) and F_(n) on c_(s). These effects are plotted in FIG. 8. Thisshows that higher sound speeds are preferable.

Listed in Table II are a variety of filling materials and their relevantproperties. For resonant driving conditions, water and 0.65 cS siliconeoil are best because of their low viscosities. Nitrogen would make apoor choice because of its very low value of static index of refraction.Because of their high viscosities, Glycerol and 100 cS silicone oil areless desirable for resonant operation.

TABLE II Properties of potential 

 fluids. All values 

 for temperatures in the 20-30° C. range. V c

p_(a) Fluid n_(s) (cS) (m/s) (kg/m³) Silicone Oil 1.4030 100 985 964Silicone Oil 1.375 0.65 873.2 761 Glycerol 1.4746 740 1904 1260 Water1.33 1.00 1493 1000 Nitrogen 1.0003 16.1 355 1.12

indicates data missing or illegible when filed

Optimizing the Driving Signal

While only sinusoidal driving signals are discussed at present, thecontroller 390 (or controller 90) can provide more complicated signalsto produce arbitrary index profiles that repeat periodically in time asdiscussed below in section II. There are two variable parameters of thesinusoidal driving signal: its amplitude, V_(A), and its frequency, ω.These two parameters will determine the inner wall velocity, which aretreated herein as a given parameter.

It has been noted that voltage amplitude, V_(A), is proportional toinner wall velocity, ν_(A). These amplitudes have a very simple effecton the refractive index. From Eqs. 38 and 39, it can be seen that lensrefractive power is directly proportional to ν_(A), and hence, V_(A),and that larger wall velocities and driving voltages are desirable.

Similar to the lens radius and sound speed, the driving frequency ω hasan effect on the refractive power of the lens 300 that cannot be givenin a simple analytic form. This effect is plotted in FIG. 9 andillustrates that lower frequencies yield greater refractive powers. Thisis because higher frequencies exhibit greater viscous damping.

Optimizing Nonresonant Conditions

There are conditions where driving on resonance is impractical. Forexample, due to the sharpness of the resonant peaks, a small error inlens properties or driving frequency can result in a large error inrefractive index. Operating off resonance can be more forgiving in termsof error, however this comes at the expense of reduced refractivepowers. In this section the off-resonant base case frequency is used.

Since the lens 300 is operating off resonance, the refractive index isgiven by Eq. 36 with n_(A), given by Eq. 38, which yields an RP_(A) of0.0016. The dependencies of RP_(A) on lens length L, static refractiveindex n₀, and driving amplitude V_(A)(ν_(A)) are all identical to whatwas found for resonant driving conditions. This is because thesevariables only appear in the common prefactors in Eqs. 38 and 39. Hencethese parameters will not be reexamined in this section. Note that ifreferring to FIG. 7 (RP_(A) vs. n₀), the RP axis will have to be scaledappropriately because the values of RP_(A) differ between the resonantand off-resonant cases.

The difference between the resonant and off-resonant driving conditionsis found in the lens radius r₀, the sound speed c_(s), and the drivingfrequency ω. These dependencies are plotted in FIGS. 10-12. Theseparameters all exhibit opposite trends from resonant driving conditions.For best off-resonant performance, large radius lenses filled with lowspeed of sound fluids driven at high frequencies are desirable. Thisoccurs because viscous damping no longer affects the refractive power.These results are expected because larger lenses vibrating at the samewall speed cause more acoustic power to be focused at the center of thelens 300, increasing refractive powers. Also, higher driving frequenciescondenses the spatial oscillations in density, producing highergradients in refractive index.

Looking at the values in Table II, it is evident that for off-resonantdriving, both silicone oils, glycerol, and water all become viable fluidchoices now that viscosity is unimportant. These fluids all haveappreciable static refractive indices compared to nitrogen. The siliconeoils are expected to have somewhat better performance over glycerol andwater because of their low sound speeds.

Other Considerations

Preventing cavitation is another consideration involved in selecting afilling material other than simply maximizing the refractive power. Ifthe pressure within the lens 300 drops below the vapor pressure of thefluid, then cavitation can occur, producing bubbles within the lens 300that disrupt its optical capability. Specifically, this can happen when

$\begin{matrix}{\mspace{79mu} {{\text{?} > {\text{?} - {\text{?}/\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (41)\end{matrix}$

where p_(v) is the vapor pressure of the fluid. There are a couple waysthat cavitation can be avoided. First, one can choose a fluid with a lowvapor pressure. Second, the lens 300 can be filled to a high staticpressure.

Another danger in blindly maximizing the refractive power is that athigh RP values, the model may break down. This is because thelinearization performed above is only valid at relatively smallamplitudes. Once the order of ρ_(A) or n_(A) becomes comparable to theorder of ρ_(o) or n₀, the linearization loses accuracy. It is likelythat the general trends observed in this section will hold to somedegree in the nonlinear regime, although the specific form of thedependence of refractive power on all the variables requires furtheranalysis. It is possible to increase the domain of the linear regime byselecting fluids of large density. One should also note that theselection of n₀ does not affect the linearization of the fluidmechanics. Therefore, increasing the refractive power via increasing thefluid's refractive index will not endanger the fluid linearization,although it may endanger the Lorentz-Lorenz linearization. However, whenthe linear models no longer apply, it is still possible to obtainsolutions via full numerical simulations.

The results of the predictive model are useful for optimizing the TAGlens design in terms of maximizing its ability to refract light insteady-state operation. A TAG lens 300 is most effective when it islong, filled with a fluid of high refractive index, and driven withlarge voltage amplitudes. If driving on resonance, lower frequencies,smaller lens radii, and fluids with larger sound speeds and lowerviscosities enhance refractive power. Off resonance, higher frequencies,larger lenses, and lower sound speeds are preferred. Viscosity isirrelevant for nonresonant driving.

It is important to note that these choices are only best for optimizingthe steady-state refractive power where the linear model is applicable.If wave amplitudes become too great, then a nonlinear model will berequired, which could be implemented numerically. Also, differentoptimization parameters will occur if, for example, one wishes tooptimize the TAG lens 300 for pattern switching speed or high damagethresholds—two of the potential advantages of TAG lenses over spatiallight modulators.

The above modeling has been done in a circular cross-section geometry soas to model a TAG lens 300 capable of generating Bessel beams. Othergeometries are also possible for creating complicated beam patterns. Thenatural example is that of a rectangular cavity, e.g., FIGS. 4A, 4B, inwhich the Bessel eigenfunctions would be replaced by sines and cosines.With other geometries that break the circular symmetry, it may also bepossible to create Laguerre-Gaussian and higher-order Bessel beams. Ithas been shown that passing a Laguerre-Gaussian beam through an axiconcreates higher-order Bessel beams. This same method can be implementedwith the TAG lens 300 replacing the axicon to produce tunable higherorder Bessel modes.

II. Determination of Voltage Signal to Create Specific Refractive IndexProfile

In the linear regime, the cylindrical TAG lens 100, 300 has thepotential to create arbitrary (non-Bessel) axisymmetric beams. Bydriving the lens 100, 300 with a Fourier series of signals at differentfrequencies, interesting refractive index distributions within the lens100, 300 can be generated. This is because the lens 100, 300 effectivelyperforms a Fourier-Bessel transform of the electrical signal into theindex pattern. As this pattern will vary periodically in both space andtime, it will be best resolved with a pulsed laser synchronized to theTAG lens 100, 300.

This section solves the inverse problem: determining what voltage signalis necessary to generate a desired refractive index profile. However,before directly tackling this question, it is easier first to find theresponse of the lens 100, 300 to a single frequency, and then to solvethe forward problem addressed in the next section: determining the indexprofile generated by a given voltage input.

The first step of the procedure is to find the frequency response of theTAG lens 100, 300. A linear model of the TAG lens 100, 300 is assumed.That is, the oscillating refractive index created within the lens 100,300 is assumed linear with respect to the driving frequency. Listedbelow are the single-frequency input signal and resulting outputrefractive index within the lens 100, 300.

$\begin{matrix}{\mspace{79mu} {{V( {\text{?},f} )} = {\text{?}\lbrack {{\hat{V}(f)}\text{?}} \rbrack}}} & (42) \\{{\text{?} = {\text{?} + {\text{?}\lbrack {\text{?}(f)\text{?}} \rbrack}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (43)\end{matrix}$

Here, f is the electrical driving frequency of the lens 100, 300, n₀ isthe static refractive index of the lens 100, 300, V(f) is the drivingvoltage complex amplitude, and k is the spatial frequency given byf/c_(s) where c_(s) is the speed of sound within the fluid.

From this frequency response, a transfer function can be defined torelate the index response to the voltage input:

$\begin{matrix}{{\Phi (f)} = {\frac{\hat{n}(f)}{\hat{V}(f)} \in {C.}}} & (44)\end{matrix}$

This transfer function can either be determined empirically, or throughmodeling, and accounts for both variations in amplitude and phase.

Forward Problem

For the forward problem, it is assumed that the lens 100, 300 is drivenwith a discrete set of frequencies at varying amplitudes and phaseshifts. One could also phrase the problem in terms of a continuous setof input frequencies, however as is seen later, the solution to thediscrete set will be more useful when dealing with the inverse problem.

An input signal of the form,

$\begin{matrix}{\mspace{79mu} {{{V(t)} = {{Re}\lbrack \text{?} \rbrack}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (45)\end{matrix}$

is assumed where each V_(m) is a given complex amplitude.

By linearity and the results of the frequency response in Eq. 43, thecorresponding refractive index in the lens is known to be,

$\begin{matrix}{\mspace{79mu} {\text{?} = {{{\text{?}\lbrack \text{?} \rbrack}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (46)\end{matrix}$

The coefficients n_(m) can be determined from the frequency response tobe,

$\begin{matrix}{\mspace{79mu} {{\text{?} = {{\Phi ( \text{?} )}\text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (47)\end{matrix}$

Eqs. 46 and 47 are the solution to the forward problem.

Inverse Problem

The inverse problem is to determine what input voltage signal, V(t), isrequired to produce a desired refractive index profile, n_(goal)(r).From Eqs. 42 and 43, it is evident that the actual refractive index is afunction of both space and time, while the input electrical drivingsignal is only a function of time. As a result, it is not possible tocreate any arbitrary index of refraction profile defined in both spaceand time, however it is possible to approximate an arbitrary spatialprofile that repeats periodically in time. It will be assumed that thearbitrary profile is centered around the static index of refraction, n₀.The deviation of the goal from n₀ is denoted as n_(goal)(r), and thefrequency with which it repeats in time as f_(rep).

The procedure is depicted as a flow chart in FIG. 13. First, n_(goal)(r)is decomposed into its spatial frequencies using a Fourier-Besseltransform. This result is then discretized so that only spatialfrequencies that are integer multiples of f_(rep)/c_(s) are included.Then, in order to write the index response in the form of Eq. 46, aninverse Fourier-Bessel transform is applied to the discrete series. Thisgives the coefficients n_(m), which are used in conjunction with thefrequency response to yield the required voltage signal in the frequencydomain. Summing over all modes provides the final answer, V(t).

The first step is to decompose the desired index profile into itsspatial frequencies using a windowed Fourier-Bessel transform because ofthe circular geometry of the lens 100, 300. Given n_(goal)(r),n_(goal)(k) can be computed as

$\begin{matrix}{{{\text{?}(k)} = {\text{?}(r)\text{?}( {2\pi \; {rk}} )2\pi \; r{r}}},{\text{?}\text{indicates text missing or illegible when filed}}} & (48)\end{matrix}$

where r₀ is the inner radius of the lens 100, 300. Depending on thedesired index goal, n_(goal)(r) this windowing may introduce undesirableGibbs phenomenon effects near the edge of the lens 100, 300 ifn_(goal)(r) does not smoothly transition to zero at r=r₀. However, thesignificance of these effects can be reduced by either modifying thegoal signal, extending the limit of integration beyond r₀, or simplyusing an optical aperture to obscure the outer region of the lens 100,300.

From inverse-transforming it is known that,

$\begin{matrix}{{\text{?}(r)\text{?}}{\text{?}\text{indicates text missing or illegible when filed}}} & (49)\end{matrix}$

However, each of the spatial frequencies will oscillate in time at itsown frequency given by f=c_(s)k. As a result, the goal pattern can onlygenerated at one point in time. If this time is t=0, then the timedependent index of refraction will be given by,

$\begin{matrix}{{\text{?}( {r,t} ){\text{?}\lbrack \text{?} \rbrack}}{\text{?}\text{indicates text missing or illegible when filed}}} & (50)\end{matrix}$

In practice, one would wish the goal index pattern to repeatperiodically in time, as opposed to achieving it only at one instant intime. Therefore, the second step of the procedure is to discretize thespatial frequencies used so that the n_(goal)(r) can be guaranteed torepeat, with temporal frequency f_(rep). This is achieved by onlyselecting spatial frequencies that are multiples of f_(rep)/c_(s). Thatis, it is assumed that,

$\begin{matrix}{\mspace{79mu} {{k \in \text{?}} = {{\text{?}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (51)\end{matrix}$

The upper limit, M, is set sufficiently large so that the contributionto n_(goal)(r) is negligible from spatial frequencies higher thanMf_(rep)/c_(s). It is also required that n_(goal)(r) and f_(rep) arechosen so that the contribution is negligible from spatial frequencieslower than f_(rep)/c_(s) and so that the discretization accuratelyapproximates the continuous function. The lower f_(rep), the moreaccurately the discretization will reflect the continuous solution,however it also means that there will be longer intervals betweenpattern repetition.

This discretization changes the integral in Eq. 50 into a sum:

$\begin{matrix}{\mspace{79mu} {{\text{?} = {\text{?}\lbrack \text{?} \rbrack}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (52)\end{matrix}$

where Δk is the spacing between spatial frequencies, in this case givenby f_(rep)/c_(s). The third step of the procedure is to compute thissum. At this point, one should compare n_(goal)(r) with n(r,0) to ensuregood agreement. If the agreement is poor, lowering f_(rep), raising M,smoothing n_(goal)(r), and continuing the integration in Eq. 48 beyondr₀ can all improve the approximation.

By comparing Eq. 52 with Eq. 46, it is evident that,

$\begin{matrix}{{\text{?} = {\text{?}\Delta \; k\text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (53)\end{matrix}$

Using the frequency response in Eq. 47 and rewriting k_(m) and Δk interms of f_(rep), this expression can be used to find the voltage signalcoefficients:

$\begin{matrix}{\mspace{79mu} {{\text{?} = \text{?}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (54)\end{matrix}$

The time domain signal is given by the same expression as Eq. 45,

$\begin{matrix}{\mspace{79mu} {{V(t)} = {{{{Re}\lbrack \text{?} \rbrack}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (55)\end{matrix}$

Eqs. 54 and 55 represent the last step and solution to the inverseproblem.

In fact, discretization is not absolutely necessary to provide thistransformation. One can analytically perform the same functionsresulting in a temporal spectrum of the voltage function. In this case,the equations presented above would replace summations and series withdefinite integrals. However, the resulting voltage function would notnecessarily repeat periodically in time.

As a theoretical, exemplary problem, a simple converging lens with aspecific focal length, l, is created, as shown in FIG. 14. Given thiscriteria, the goal is to determine the voltage signal to be supplied tothe TAG lens 100, 300 so that it adopts this converging lensconfiguration with temporal period t_(rep). The parameters for the TAGlens 100, 300 in this example are given in Table 3. For the sake ofsimplicity in illustrating the above procedure, a constant frequencyresponse, Φ(f), is assumed. Note that in reality, Φ(f) would varygreatly with frequency near resonances within the lens 100, 300.However, since this affects only the last step of the procedure, theexact form of the function, while important for practicalimplementation, is unimportant here in demonstrating the solutionprocess.

TABLE 3 Parameters used in the example inverse problem. Name SymbolValue Focal Length l 1 m Temporal Period t

1 ms Inner Radius r_(s) 5 cm Lens Length L 5 cm Sound Speed c

1000

⁻¹ Transfer Func. Φ(f) 10⁻³ V⁻¹ Largest Mode M 300

indicates data missing or illegible when filed

The first step of this example is to determine the refractive indexprofile for a simple converging lens with this focal length. Using smallangle approximations, the angle by which normally incident incoming raysshould be deflected is given by,

$\begin{matrix}{\mspace{79mu} {{\theta \text{?}(r)} = {{- r}{\text{?}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (56)\end{matrix}$

This goal angular deflection is shown in FIG. 15.

The corresponding refractive index profile required to deflect rays bythis angle is given by,

$\begin{matrix}{{\text{?} = {\text{?} = {- \text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (57)\end{matrix}$

This function is plotted in FIG. 16. If the effective converging lenswere to have a high numerical aperture, were optically thick, or wereused in a situation where the small angle approximations were notaccurate enough, then the goal refractive index would no longer beparabolic. However, the TAG lens could still be used to emulate the newgoal function.

The procedure described above is now followed to obtain V(t). Equation48 is used to decompose the goal refractive index into its spatialfrequencies. These spatial frequencies are then discretized with thelower bound and spacing between frequencies given by f_(rep)=1/t_(rep)=1kHz. The upper bound is chosen to be 300 kHz, which corresponds toM=300. Both the continuous and discretized spatial frequencies areplotted in FIG. 17. The value of M was chosen large enough so that theactual angular deflection reasonably approximates the goal, as shown inFIG. 15.

The actual index of refraction at time t=0 is obtained from thediscretized frequencies using Eq. 52 and is plotted in FIG. 16. Note thevery good agreement between the goal and actual refractive indexprofiles. This good agreement is due to f_(rep) being relatively small,and M being relatively large.

The angular deflection is obtained by differentiating the refractiveindex profile. This deflection is plotted in FIG. 15. Note that thediscrepancy between the goal and actual angular deflections is amplifiedrelative to the refractive index profile. This is because of theamplifying properties of the derivative. However, there is still goodagreement between the goal and actual deflections, except at the edge ofthe lens 100, 300 where the Gibbs phenomenon can be observed because therefractive index does not smoothly transition to zero there. Betteragreement could be achieved by choosing a larger M or smoothing out thedesired index goal. In some situations, choosing a larger t_(rep) wouldalso be helpful.

Using Eqs. 54 and 55, the actual voltage signal to be generated by thecontroller 90, 390 can be computed. It is periodic with period t_(rep).One period is plotted in FIG. 18. It is important to note that changesin Φ(f) can change the form of this function. This signal can be outputfrom the controller 90, 390 to drive the lens 100, 300.

The above method to approximately generate arbitrary axisymmetric indexpatterns which repeat at regular intervals allows a cylindrical TAG lens100, 300 to act as an axisymmetric spatial light modulator. If insteadof a cylindrical geometry, a rectangular, triangular, hexagonal or othergeometry were used for the TAG lens 200 with two or more piezoelectricactuators 210, then arbitrary two dimensional patterns may beapproximated without the axisymmetric limitation. The only mathematicaldifference will be the use of Fourier transforms instead of usingFourier-Bessel transforms to determine the spatial frequencies. As such,the TAG lenses 100, 200, 300 may be used as a tunable phase mask (orhologram generator) or the adaptive element in a wavefront correctionscheme.

Compared to nematic liquid crystal SLMs, TAG lenses 100, 300 can havemuch faster frame rates limited only by the liquid viscosity and soundspeed. The frame rate of the theoretical example presented above was 1kHz. If the voltage signal was not precisely periodic in time, butvaried slightly with each repetition, then pattern variations could beachieved at this rate. This method used only steady state modeling,however with fully transient modeling even higher frame rates would bepossible. Because of the simplicity and flexibility in the opticalmaterials used in a TAG lens 100, 300, it is possible to design one towithstand extremely large incident laser energies. Due to its analognature, TAG lenses also avoid pixilation issues.

In addition to their advantages, TAG lenses do have some limitationsthat may make SLMs more suitable in certain applications. Specifically,TAG lenses may work best when illuminated periodically with a small dutycycle, whereas SLMs are “always-on” devices. In some special cases,continuous wave illumination of TAG lenses may be acceptable if theindex pattern in the middle of the cycle is not disruptive. Moreover,TAG lenses may be operated in modes other than that of a simple positivelens with fixed focal length, for example, in modes where multiscaleBessel beams are created.

III. Optical Analysis of Multiscale Bessel Beams

In this section multiscale Bessel beams are analyzed which are createdusing the TAG lens 300 of FIGS. 3A, 3B as a rapidly switchable device.The shape of the beams and their nondiffracting and self-healingcharacteristics are studied experimentally and explained theoreticallyusing both geometric and Fourier optics. The spatially and temporallyvarying refractive index within the TAG lens 300 leading to the observedtunable Bessel beams are explained. As discussed below, experimentsdemonstrate the existence of rings, and the physical theory (geometricand diffractive) accurately predicts their locations. By adjusting theelectrical driving signal, one can tune the ring spacings, the size ofthe central spot, and the working distance of the lens 300. The resultspresented here will enable researchers to employ dynamic Bessel beamsgenerated by TAG lenses. In addition, this section discusses in detailhow to tune the electrical driving signal to alter the observedpatterns.

Experimentally, it has been observed that the TAG beam has bright majorrings that may each be surrounded by multiple minor rings, depending ondriving conditions (see FIG. 19). On the large scale, the periodicity ofthe major rings is that of the square of a Bessel function. When drivenat a sufficiently high amplitude, minor rings become evident. Theperiodicity of the minor rings around the central spot is also given bythe square of a Bessel function. Therefore, the TAG lens 300 beam can bethought of as a multiscale Bessel beam, i.e., one having at least onebright major ring surrounded by at least one minor ring. The images ofFIGS. 19A and 19B are both taken 50 cm behind the lens 300 with drivingfrequency 257.0 kHz.

As shown in FIG. 34, the density variations in the standing wave resultin refractive-index variations that focus the light passing through thelens 300. The acoustic standing wave is created by an alternatingvoltage, typically in the radio-frequency range, applied to thepiezoelectric element 310 by the controller 390. The relations betweenthe drive frequency and amplitude, and the resulting refractive-indexmodulation, are nonlinear and complex, but are predictable as shown insection I.

Although the refractive-index variation induced by the voltage is small,the lens 100, 200, 300 is thick enough to allow significant focusing.Because the index variation is periodic, the TAG lens 100, 300 is ablenot only to shape a single beam of light, but in a rectangularconfiguration the TAG lens 220 can also take a single beam and create anarray of smaller beams as do other adaptive optical devices.

The optical properties of the lens 300 are determined by a number ofexperimentally controllable variables. First and foremost, the geometryand symmetry of the lens 300 determine the symmetries of the patternsthat can be established. For instance, a square shaped lens 200 canproduce a square array of beamlets, while a cylindrical geometry canproduce Bessel-like patterns of light. The density and viscosity of therefractive fluid, the static filling pressure, as well as the type ofpiezoelectric material, will all play a role in determining the staticand dynamic optical properties of the TAG lens 300. These properties canbe designed and optimized for different applications.

The two “knobs” that control the TAG lens 300 effect on a continuous(CW) beam are the amplitude and frequency of the electrical signalapplied to the piezoelectric transducer 310. For a pulsed beam ordetector, extra control is provided by the timing of pulse, as discussedin section V. The amplitude determines the volume of the sound wave andthe corresponding amplitude of the index variation. Of course, there arefundamental limitations to how much index amplitude is possible due tothe relatively small compressibility of liquids. However, the lens 300does not need to be filled with a liquid at all. The physics of thisdevice should work equally well on a gas, solid, plasma, or amulticomponent, complex material. The frequency of the drive signaldetermines the location of the maxima and minima in the index function.Multiple driving frequencies and segmented piezoelectric elements can beused to give full functionality to this device and enable creation ofarbitrary patterns.

FIG. 20A depicts the experimental setup. The TAG lens 300 is primarilystudied by illuminating it with a wide Gaussian beam of collimated 532nm CW laser light. (Although in the next section the effects of pulsedillumination are discussed.) The intensity pattern produced by the lens300 is then sampled at various distances using a ½″ CCD camera 370 (Cohu2622). In order to achieve intensity profiles as a function of radius,azimuthal averaging was used to filter out CCD noise. In order toobserve the time dependence of the beam, a pulsed 355 nm laser with 20ns pulse length is also used to strobe the pattern.

The TAG lens 300 has an inner diameter of 7.1 cm and a length of 4.1 cmincluding the piezoelectric element 310, contact ring 320, and gaskets350, 352. The fluid used is 0.65 cS Dow Corning 200 Fluid (siliconeoil), which has an index of refraction of n₀=1.375, and speed of soundof 873 ms⁻¹ under standard conditions. The TAG lens 300 is driven by acontroller 390 comprising a function generator (Stanford ResearchSystems, DS345) passed through an RF amplifier (T&C Power Conversion, AG1006). An impedance-matching circuit of the controller 390 is used tomatch the impedance of the TAG lens 300 at its operating frequencieswith the 50Ω output impedance of the RF amplifier. A fixed componentimpedance matching circuit is used, which works well over the range 100kHz-500 kHz. Most of the data presented here is acquired at a frequencyof 257.0 kHz. If driven near an acoustic resonance of the lens 300, thenthe amplifier and impedance matching circuit are unnecessary, and thismodified setup has been used to acquire data over larger frequencyranges. The data presented herein cover the range 250 kHz-500 kHz atamplitudes from 0-100 V peak-to-peak.

The driving parameters were chosen to best illustrate the multiscalenature of the Bessel beam. The TAG lens frequencies are chosen so thatthe lens 300 appears to be operating close to a single-mode resonance.Driving amplitude was chosen to provide well-defined major and minorrings for this example. The imaging distance for fixed-z figures waschosen to be approximately at the midpoint of the multi-scale Besselbeam.

The coordinate system used for presenting the theoretical calculationsand experimental results is defined with z in the direction of thepropagation of the light, x and y being transverse coordinates at theimage plane, and ξ and η being transverse coordinates at the lens plane,as shown in FIG. 20B. The radial coordinates are given by

$\mspace{85mu} {\text{?} = {{\sqrt{\text{?} + \text{?}}\mspace{14mu} {and}\mspace{14mu} \rho} = {{\sqrt{\text{?} + \text{?}}.\text{?}}\text{indicates text missing or illegible when filed}}}}$

Theory and Numerical Methods

The ultimate goal of the following theory is to describe the physics oflight propagation through the lens 300, particularly in the case whencoherent, collimated light is shone through it.

Refractive Index Profile

The first step in modeling the TAG lens 300 is to determine the index ofrefraction profile. It has been calculated in section I above that therefractive index within the TAG lens 300 is of the form,

$\begin{matrix}{\mspace{79mu} {{\text{?} = \text{?}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (58)\end{matrix}$

assuming a low viscosity filling fluid (kinematic viscosity much lessthan the speed of sound squared divided by the driving frequency) withlinearized fluid mechanics, where n₀ is the static index of refractionof the filling fluid, ω is the driving frequency of the lens 300, c_(s)is the speed of sound of the filling fluid, and ρ is the radialcoordinate in the lens plane. This function is plotted in FIGS. 21A and21B, at two different times. If driven on resonance, then the sinechanges to a cosine, however as this section focuses on the standingwaves within the cavity, and not on transient effects, the temporalphase shift is irrelevant.

The only parameter in Eq. 58 with some uncertainty is n_(A). Themodeling in section I estimates its value, however n_(A) is verysensitive to a number of experimental parameters, most notably how closethe driving frequency is to a resonance. Because of this highsensitivity and experimental uncertainty in some of the modelingparameters, here n_(A) is treated as a fitting parameter, adjusting itsvalue in order to achieve the best agreement between the theory and theexperiments. This results in values for n_(A) on the order of 10⁻⁵ to10⁻⁴, in good agreement with modeling predictions of section I above.

It is important to note that the refractive index is a standing wavethat oscillates in time. This time-dependent index is illustrated inFIGS. 21A and 21B. The theoretical predictions for the correspondingtime-dependent patterns are presented in FIGS. 21C and 21D. Experimentalimages are also presented for comparison in FIGS. 21E and 21F. Theexperimental images were acquired by operating a pulsed 20 ns laser at arepetition rate synchronized to the TAG lens driving frequency. Thissynchronization was implemented so that the laser fired at the samerelative phase in each period of the TAG lens oscillation, however thelaser did not fire on every TAG period, resulting in effective laserrepetition rates below 1 kHz. By adjusting the relative phase betweenthe TAG lens 300 and the laser, one can shift the pattern from FIG. 21Eto FIG. 21F and back to FIG. 21E. The phase delay between these twopatterns corresponds to half the period of the driving signal of the TAGlens 300. As expected, one sees bright regions at local maxima in therefractive index, and dark regions at local minima. This time-varyingnature of the beam presents interesting opportunities for generatingannular patterns in pulsed-laser applications. The theoretical andexperimental CW patterns presented in the remainder of this section area time-average of the intensity patterns resulting from both upward anddownward pointing index profiles (and the continuously varyingintermediate profiles). As a result, major scale rings are observed withBessel-squared periodicity.

Geometric Optics

The TAG lens 300 is modeled using the thin lens approximation, that is,a light ray exits the lens 300 at the same transverse location where itentered the lens 300. The conditions where this approximation is validare examined below. Under the thin lens approximation, the phasetransformation for light passing through a lens 300 is given by:

$\begin{matrix}{\mspace{79mu} {{t\text{?}( {\xi,\eta} )} = {{{\exp ( {\text{?}( {{nL} + {L\text{?}} - L} )} )}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (59)\end{matrix}$

where k₀ is the free-space propagation constant (2π/λ), L₀ is themaximum thickness of the lens 300, L is the thickness at any given pointin the lens 300, and n is the index of refraction at any given point inthe lens 300. Since the lens 300 is a gradient index lens, L=L₀throughout the lens 300, and it is only n that varies transversally.

In some cases, it may be useful to express the phase transformation inEq. 59 as the angle at which a collimated light ray would leave the lens300. After having traveled through the bulk of the lens 300, but justbefore exiting, the equation for the wavefront is k₀n(ρ)(L₀+z)=const. Atthis point, an incident light ray would have been deflected by an angle,θ, that is perpendicular to the wavefront and is hence given by:

$\begin{matrix}{\mspace{79mu} {{{\tan ( \text{?} )} = {{- \frac{\text{?}}{\text{?}}}\frac{{n(\rho)}}{\rho}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (60)\end{matrix}$

assuming the thin lens approximation. To get the angle of a light rayleaving the lens 300, Snell's law is applied to the fluid-air interface(since the transparent window 330 of the lens 300 is flat, it has noeffect on the angle of an exiting ray). This yields θ, the angle that aray will propagate after leaving the lens 300.

$\begin{matrix}{\mspace{79mu} {{{\sin ( {\theta (\rho)} )} = {{n(\rho)}{\sin ( {\text{?}(\rho)} )}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (61)\end{matrix}$

assuming that n(ρ)/n_(air)≈n(ρ) . . . . Applying small angleapproximations to Eqs. 60 and 61 yields

$\begin{matrix}{{\theta (\rho)} = {{- L_{o}}{\frac{n}{\rho}.}}} & (62)\end{matrix}$

In order to illustrate the physics behind the minor ring interferencepatterns created by the TAG lens 300, it is useful to consider a linearapproximation to one of the peaks, as is shown in FIG. 21A. In thiscase, linearization about the inflection point of the central peak hasbeen performed. A gradient index of refraction lens 300 with this linearprofile is fundamentally equivalent to a uniform-index conical axicon.As long as the input beam completely covers the central peak, thisapproximation is valid and reproduces the key elements of the observedbeam. However, if one apertured the lens 300 so that the input beam onlycovered the rounded tip of the central peak, then the linear/axiconapproximation would not be sufficient, and a second-order parabolicapproximation would be required.

The equivalence between the TAG lens 300 and an axicon can be shownquantitatively. To determine the angle that a light ray leaves anaxicon, Snell's law is used:

$\begin{matrix}{\mspace{79mu} {{\sin ( {\phi + {\theta \text{?}}} )} = {n\text{?}\sin {\text{?}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (63)\end{matrix}$

Here, θ_(ax) is the angle from the z-axis that a light ray leaves theaxicon, and φ is the angle between the z-axis and the normal to theoutput face of the axicon. The cone angle of this axicon is given byα=π−2φ. Substituting in for φ from Eq. 63, setting θ_(ax)=θ from Eq. 62,and applying small angle approximations, it is possible to express thecone angle of the corresponding axicon in terms of the parameters of thelinear gradient in index of refraction lens:

$\begin{matrix}{\mspace{79mu} {\text{?} = {{\pi - {2\; \phi}} = {\pi - {2{{{\text{?}\frac{n}{\rho}}}.\text{?}}\text{indicates text missing or illegible when filed}}}}}} & (64)\end{matrix}$

This equation forms the basis for the effects of tuning the lens 300 bychanging the driving amplitude. Increasing the driving amplitudeincreases dn/dρ and is therefore identical to increasing the cone angleof the equivalent axicon.

Even though this model of the TAG lens 300 ignores the curvature of therefractive index, it does a good job of qualitatively explaining thevisible features, and furthermore the experimental pattern around thecentral spot does closely resemble Bessel beams generated by axicons, asshown below in the section labeled “Beam characteristics”. This modelhas so far neglected the time dependence of the refractive index.However, simulations show that the periodicity of the time-averagepattern surrounding the central spot is closely approximated by theinstantaneous pattern produced when the refractive index at the centerof the lens 300 is at its peak. While not significantly shifting theirpositions, the time-averaging does decrease the contrast of the minorrings.

The major rings and their surrounding minor rings can be explained in asimilar way. The only difference between these minor rings and thosesurrounding the central spot is that these rings are derived fromcircular ridges in the refractive index as opposed to a single peak. Forexample, the first major ring is established from the peaks highlightedin FIG. 21B, a half-period later in time with respect to the centralspot from FIG. 21A. Due to the time-averaging CW nature of the imagingmethod, images such as FIG. 19 exhibit major rings at the locations ofthe peaks in FIG. 21A, as well as the peaks in FIG. 21B. The result isthat the major ring locations have a Bessel-squared periodicity—the sameperiodicity as conventional Bessel beams.

Fourier Optics

Using the phase transformation for light passing through the lens 300given by Eq. 59, the electric field of the light upon exiting the lens300 is given by:

$\begin{matrix}{\mspace{79mu} {{{U\text{?}( {\xi,\eta} )} = {t\text{?}( {\xi,\eta} )U\text{?}( {\xi,\eta} )}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (65)\end{matrix}$

where U₀(ξ,η) is the electric field of the light entering the lens 300.

In order to find the intensity profile at the image plane, the fieldU_(TAG)(ξ,η) must be propagated using a diffraction integral. TheRayleigh-Sommerfeld diffraction integral is used in this simulation. Theassumptions involved in this integral are that the observation point ismany wavelengths away from the lens 300 (r>>λ) and the commonly acceptedassumptions of all scalar diffraction theories. The field at a distancez from the lens plane is given by:

$\begin{matrix}{\mspace{79mu} {{{\text{?}( {x,y,z} )} = {\text{?}( {\xi,\eta} )\frac{\text{?}}{\text{?}}\text{?}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (66)\end{matrix}$

where the integration is performed over the entire aperture of the lens300 and s(x, y, ξ, η) is the distance between a point (ι, η) on the lensplane and a point (x, y) on the image plane, given by:

$\begin{matrix}{\mspace{79mu} {{\text{?} = \sqrt{\text{?} + {( {x - \xi} )\text{?}} + {( {y - \eta} )\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (67)\end{matrix}$

Computationally, the integral in Eq. 66 can be difficult to evaluatebecause the magnitude of k₀ (on the order of 10⁷ m⁻¹) leads to a phasefactor in the integrand that varies rapidly in ξ and η. Numericalapproximations therefore require a sufficient number of points in thetransverse directions to accurately represent the variation of thisphase factor in the domain of interest. The closer the image plane tothe lens plane, the more quickly that phase factor will vary (because sbecomes more strongly dependent on ξ and η now that z is small), and themore points are required to accurately compute the integral. Note thatthe integral is a convolution as the integrand is a product of twofunctions, one of (ξ, η) and another of (x−ξ, y−η). In this study theconvolution integral is computed using fast Fourier transforms (FFTs).

Finally, the intensity profile at the image plane is found from theelectric field as follows:

$\begin{matrix}{\mspace{79mu} {{{\text{?}( {x,y,z} )} = {\frac{1}{2}\sqrt{\text{?}}{{\text{?}( {x,y,z} )}}\text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & {(68)\;}\end{matrix}$

In the following section, all theoretical figures are obtained usingthis Fourier method, assuming a refractive index of the form of Eq. 58and averaging many images corresponding to instantaneous patternsgenerated at different times within one period of oscillation.

Beam Characteristics

This section describes the characteristics of the TAG-generatedtime-average multiscale Bessel beam. The TAG beam characteristics aredivided into two categories: the nature of the beam propagation and theability to tune the beam. In each category, theoretical predictions,experimental results, and comparisons between the two are presented. Thespecific propagation characteristics are the beam profile, the axialintensity variation, the beam's nondiffracting nature, and the beam'sself-healing nature. The parameters of the beam that are tunable includethe major ring locations, the minor rings locations, the central spotsize, and the working distance. Intensity values in all plots (excepton-axis intensity) have been normalized so that the wide Gaussian beam(full width at half-maximum greater than 1 cm) incident to the TAG lens300 has a peak intensity of 1.

Beam Propagation

The first important demonstration is that the lens 300 does produce amultiscale Bessel beam. FIG. 19 shows the multi-scale nature of theintensity pattern, while FIG. 22 shows how the minor scale and firstmajor ring of the experimental TAG beam propagate. Note that the centrallobe of the beam propagates over a meter without significantdiffraction—one of the key properties of a Bessel beam. No experimentalmethod of creating an approximate Bessel beam creates a trulynondiffracting beam because all experimental methods limit beams withfinite apertures, whereas a true Bessel beam is infinite in transverseextent.

An intensity profile of the multiscale Bessel beam is plotted in FIG.23. This is a slice of FIG. 22, 70 cm behind the lens 300. One canobserve four minor rings with radius less than 1 mm. The location ofthese rings is consistent between the experimental and theoreticalcurves, however the peak intensity is significantly lower in theexperiment, most likely due to wavefront errors either from the incidentbeam or from scattering within the TAG lens fluid or windows. Anotherpossible explanation is a discrepancy between Eq. 58 and the realsystem, perhaps because of nonlinear acoustic effects within the lens300, or because of the excitation of higher order acoustic Bessel modes(J_(n)( ),n>0) because of slight nonuniformities in the piezoelectric.Despite this error in peak intensity, good theoretical-experimentalagreement was found in the minor ring location by fitting n_(A) from Eq.58 to the experimental data. The resulting value of n_(A) is 4×10⁻⁵.Neglecting the intensity difference, the following figures will showthat this single parameter fit accurately describes all the spatialcharacteristics of the observed TAG beam. Note that the location of thefirst major ring is accurately predicted by the model.

The minor ring fringes in FIG. 23 are similar to the fringes one wouldexpect from an axicon. Performing the linear approximation illustratedin FIG. 21A, and using Eq. 64, the cone angle of the correspondingaxicon is 179.5°. For many applications, cone angles close to 180° areused because of their long working distance and large ring spacing.Experimentally so far, equivalent cone angles as sharp as 178° have beenachieved, however sharper cone angles can be achieved by increasing thethickness of the lens 300 (L₀) or further increasing the drivingamplitude, which are both linearly proportional to the range of coneangles (Eq. 58). In addition, optimizing the driving frequency can leadto sharper effective cone angles by taking advantage of the acousticresonances at certain frequencies.

The essential characteristic of nondiffracting beams is that thetransverse dimensions of the central lobe remain relatively constant inz. From FIG. 22, one can see that this is the case for the TAG beam.This property is plotted quantitatively in FIG. 24. Although the TAGbeam does diverge in both theory and experiment because it is not anexact Bessel beam, this divergence is small compared to that of aGaussian beam, and similar to that of an axicon-generated beam.

If a conventional collimated Gaussian beam is focused so that it has aminimum spot size (radius at which the electric field amplitude falls to1/e of its peak value) of 150 μm at z=58 cm, then by the time the lightreaches z=100 cm, the spot size will be more than 500 μm. In contrast, aTAG beam with this beam waist would only diverge to a size of 175 μmafter this distance. This Gaussian beam waist is chosen to match theexperimental width of the TAG beam at the location where the theoreticalTAG beam reaches its peak intensity.

Apart from being nondiffracting, the other major feature of Bessel beamsis their ability to self-heal. Because the wavevectors of the beam areconical and not parallel to the apparent propagation direction of thecentral lobe of the beam, the intensity pattern of Bessel beams iscapable of reforming behind obstacles placed on-axis. This feature isexperimentally demonstrated for a TAG beam in FIG. 25. An obstruction isplaced slightly after the initial formation of the TAG minor scaleBessel beam. This obscures the beam for a short distance, however theBessel beam eventually heals itself and reforms approximately 30 cmbeyond the obstruction.

Tunability

One of the most innovative features about the TAG lens 300 is theability to control the shape of the emitted beam. One can directly tunethe major and minor ring sizes and spacings without physically movingany optical components. The major and minor scales of the Bessel beamare both adjustable because of the two degrees of freedom in thetime-average pattern: the driving frequency and the driving amplitude.Changing the driving amplitude modifies only the minor scale, whilechanging the driving frequency excites different cavity modes and willaffect both scales of the beam. Independent tunability of the major andminor rings is useful in applications such as optical tweezing formanipulating trapped particles relative to each other, laser materialsprocessing for fabricating features of different size, and scanning beammicroscopy for switching between high-speed coarse images and slow-speedhigh-resolution images.

Tuning the major ring spacing can be achieved by modulating the drivingfrequency as shown in section I above. This is because the major ringsoccur near the extrema of J₀(ωρ/c_(s)) from Eq. 58. Increasing thedriving frequency compresses the major rings, while decreasing thedriving frequency increases their spacing. The radial coordinate of thefirst major ring, ρ*, is approximately given by

$\begin{matrix}{\mspace{79mu} {{\text{?} = \frac{\text{?}}{\omega}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (69)\end{matrix}$

where 3.832 is the radial coordinate of the first minimum of J₀(ρ). Thisfunction is plotted in FIG. 26, along with experimental measurements.Optical propagation slightly shifts the theoretical position of theintensity maximum relative to the index maximum because the refractiveindex profile is not locally symmetric between the inside and theoutside of the first major ring. The experimental results agree closelyto the predictions, however they do exhibit some variability betweentrials. This is attributed to different filling conditions of theprototype lens 300. When refilling the lens 300 between trials, smallchanges in the pattern are noticed, and also some azimuthal asymmetries.Any asymmetries represent the contribution of non-J₀ modes within thelens 300. These modes exhibit different radial index distributions andtherefore result in slight shifting of the first major ring. The effectsof changing the driving frequency on the minor scale pattern arecomplicated due to resonant amplitude enhancement at certain drivingfrequencies.

The continuous tunability of the minor Bessel rings is demonstratedexperimentally (FIGS. 19A, 19B, 27A, 33A, 33B) and theoretically (FIG.27B) by varying the driving amplitude of the TAG lens 300. Therefractive index amplitude, n_(A), is directly proportional to thedriving voltage amplitude. This conclusion is qualitatively supported bythe similarity between FIGS. 27A and 27B. The theoretical change in thepattern with increasing index amplitude closely resembles theexperimentally observed change in the pattern with increased voltageamplitude.

Increasing the driving amplitude increases the number of discernableminor rings (FIGS. 27A, 27B, 33B) and decreases the spacing betweenthose rings (FIGS. 28, 33A) because it alters the angle that light rayssurrounding a peak are deflected, similar to sharpening the cone angleof an axicon and increasing the spatial frequency of the resultinginterference fringes. (Note in FIGS. 27A and 27B the gray-scale map hasbeen scaled down for clarity. The actual peak intensity is 51.) Therange of plotted refractive index amplitudes corresponds (at thisfrequency) to equivalent cone angles from 180° (lens off) to 179°.Eventually, a large enough amplitude results in interference between theminor rings surrounding adjacent major rings. Driving amplitude does notaffect the major scale beam because it does not alter which acousticmodes are excited within the lens 300. Such changes directly correspondto changing the axicon's cone angle.

The driving amplitude can also be used to tune the working distance.Here, the working distance is defined as the smallest distance behindthe TAG lens 300 where a minor ring is observed surrounding the centralpeak (the distance until the start of the Bessel beam). This can beinferred from FIGS. 27A and 27B. At low amplitudes, the Besselinterference pattern is not apparent, implying that the working distanceat these amplitudes is greater than 50 cm, the distance behind the lens300 that these plots were obtained. As the driving amplitude increases,the working distance decreases, and the multiscale Bessel beam formsnearer the lens 300. This ability to tune working distance should beespecially useful for dynamic metrological applications, such as thoseinvolved in long range straightness measurements. The TAG lens 300 wouldbe helpful for using a single optical setup to very quickly switchbetween straightness measurements on the order of centimeters tostraightness measurements on the order of hundreds of meters. Inaddition, being able to tune the long working distance of the Besselbeam should prove useful for scanning beam microscopy.

This section has modeled and experimentally characterized TAG-generatedmultiscale Bessel beams. This characterization has verified therefractive index model for the TAG lens 300 presented earlier. Inaddition, the connection between the minor scale of the TAG beam andrefractive axicons has been established. The nondiffracting andself-healing characteristics of the TAG lens beam have beenexperimentally proven and theoretically justified. The ability toindependently tune the major and minor scales of the beam throughdriving frequency and amplitude has also been presented, along with thetunability of the central spot size and working distance.

Because of this tunability, TAG lenses may be used in applications wheredynamic Bessel beam shaping is required. In particular, applicationsinclude optical micromanipulation, laser-materials processing, scanningbeam microscopy, and metrology and others.

If a similar analysis is performed for a TAG lens 200 with a squarechamber, then instantaneous patterns such as those in FIG. 43 would bepredicted, while time average patterns would look like those shown inFIG. 44. Indeed similar time-average patterns are seen experimentally(FIG. 42), if without the fine detail, due to system aberrations. Onesees that this pattern is similar to that created by a square array oflenslets.

Justification for Approximation that TAG Lens is a Thin Lens

The TAG lens 300 can be approximated as a thin lens. This approximationis valid if a ray of light does not significantly deflect while passingthrough the lens 300. A corollary to this is that a ray experiences aconstant transverse gradient in index of refraction throughout itstravel within the lens 300. A specific definition for what it means fora deflection to be “significant” is provided below.

A bounding argument is used to justify the thin lens approximation. Itcan be shown that the actual transverse deflection of a ray passingthrough the lens 300 is less than the deflection predicted by a thinlens model, which is in turn much less than the characteristictransverse length scale: the spatial wavelength of the acoustic Besselmodes within the lens 300. This length scale is chosen because itimplies that the deflected ray experiences a relatively constantgradient in refractive index while passing through the lens 300:

Let |δ|_(max) be the maximum deflection experienced by a ray. If thelens 300 is thin then the corresponding exit angle of the ray (beforethe fluid-window interface) is given by θ(ρ) from Eq. 60, reproducedhere:

$\begin{matrix}{\mspace{79mu} {{\tan ( {\text{?}(\rho)} )} = {{- \frac{\text{?}}{n(\rho)}}{\frac{{n(\rho)}}{\rho}.\text{?}}\text{indicates text missing or illegible when filed}}}} & ({A1})\end{matrix}$

In the thin lens case, there is a one-to-one correspondence between theray with the largest exit angle, θ_(max) and the ray with the largestdeflection, |δ|_(thin,max). Furthermore, from Eq. A1, one can see thatthis ray passes through the region of the lens 300 with the greatestgradient in refractive index, (dn/dρ)_(max).

If the lens 300 was not a thin lens, light rays passing through it woulddeflect and therefore a single ray could not experience a refractivegradient of (dn/dρ)_(max) during its entire trip. In fact, in someregions it must experience smaller gradients, and hence the deflectionof a ray passing through the same lens 300 with thick lens modelingwould actually be smaller than calculated with thin lens modeling. Thedeflection predicted under the thin lens model, |δ|_(hrmthin,max),therefore serves as an upper bound for the true deflection, |δ|_(max).

Going back to the thin lens model, one can bound the transversedeflection based on the exit angle. The angle of the light raymonotonically increases as it passes through the lens 300 until itreaches the value of the exit angle. Hence, the total deflection, |δ|,must be less than the length of the lens 300 times the tangent of theexit angle.

Putting all of the above inequalities together, one can now bound thetotal deflection in the lens 300 assuming a thick lens model:

$\begin{matrix}{\mspace{79mu} {{\text{?} = \text{?}}{\text{?}\text{indicates text missing or illegible when filed}}}} & ({A2})\end{matrix}$

One can easily compute the value of the term on the right hand side ofEq. A2. If this turns out to be much less than the spatial wavelength ofthe acoustic fluid mode, then one can conclude that the TAG lens 300 isa thin lens. For the patterns studied here, the spatial wavelength ofthe acoustic mode is approximately 3 mm. For n_(A)=4×10⁻⁵ (best fitvalue from experimental data), the maximum gradient in index ofrefraction is 0.043 m⁻¹. Eq. A2 then gives |δ|_(max)<50 μm, which ismuch less than 3 mm. Therefore the thin lens approximations madethroughout this section are justified, and one can build on the resultsof this section to exploit TAG lenses to effect dynamic pulsed-beamshaping.

IV. Dynamic Pulsed-Beam Shaping

The ability to dynamically shape the spatial intensity profile of anincident laser beam enables new ways to modify and structure surfacesthrough pulsed laser processing. In one of its aspects, the presentinvention provides a device and method for generating doughnut-shapedBessel beams from an input Gaussian source. The TAG lens 100, 300 iscapable of modulating between focused beams and annular rings ofvariable size, using sinusoidal driving frequencies. Lasermicromachining may be accomplished by synchronizing the TAG lens to a355 nm pulsed nanosecond laser. Results in polyimide demonstrate theability to generate adjacent surface features with different shapes andsizes.

The experimental setup used for dynamic pulsed-beam shaping is shown inFIG. 29. The light source is a Nd:YVO₄ laser 362 (Coherent AVIA)delivering 15 ns duration pulses at a 355 nm wavelength and maximumrepetition rate of 250 kHz. The output beam has a Gaussian profile, witha measured diameter of about 3.5 mm at 1/e² and an M² approximately 1.3.The beam is directed into the 3.5 cm radius TAG lens 300 of FIGS. 3A, 3Bfilled with silicone oil (0.65 cS Dow Corning 200 Fluids). Thecontroller 390 comprising a wavefunction generator (Stanford ResearchSystems Model DS 345) provides a radio frequency (RF) sinusoidal signalbetween 0.33 and 1.20 MHz to drive the piezoelectric crystal, generatingvibrations inside the silicone oil, though other signals than a singlefrequency sinusoid may be used, such as a sum of two or more sinusoidsof differing frequency, or a Fourier series per Eqs. 54 and 55, forexample. The refractive index of the TAG lens cavity continuouslychanges with the instantaneous value of the AC signal.

Synchronization of the laser 362 and TAG lens 300 is accomplished usinga pulse delay generator triggered off the same AC signal. A pulse delaygenerator (Stanford Research Systems Model DG 535), which may beprovided as part of the controller 390, is programmed to provide aspecific phase shift from the trigger signal that can be much greaterthan 2ρ. In this way, it is possible to synchronize the laser pulseswith the TAG lens 300 so that each pulse meets the lens 300 in the samestate of vibration. Because the phase shift is greater than one period,the effective repetition rate of the laser pulse can be arbitrarilycontrolled within the specifications of the laser source.

For micromachining, the size of the shaped laser beam is reduced by apair of lenses, L₁, L₂, with focal lengths of 500 mm and 6 mmrespectively. The demagnified laser beam illuminates the surface of athin layer (about 4.7 μm) of polyimide coated on a glass plate 380,which is mounted on an x-y-z translation system. Photomodified samplesare then observed under an optical microscope and characterized byprofilometry.

Results, Instantaneous Patterns (Basis)

There are three main parameters of the TAG lens 300 that affect thedimensions and shapes of the patterns that can be generated: the drivingamplitude, the driving frequency, and the phase shift between thedriving signal and the laser trigger. For this section, attentionfocuses on simple shapes including annuli and single spots, although theTAG lens 300 is capable of more complex patterns. These instantaneouspatterns are denoted herein as the “basis”. In general, the frequencyaffects the diameter of the ring, the amplitude affects the sharpnessand width of the rings. The phase selects the nature of theinstantaneous pattern. For instance, when the index of refraction is ata global maximum in the center, the instantaneous pattern is a spot, butat half a period later when it becomes a global minimum, theinstantaneous pattern is doughnut or annular shaped.

In FIG. 30, a few elements of the basis are presented. These patternsare acquired 50 cm away from the TAG lens 300 but without the reducingtelescope lenses L₁, L₂. Rings with various diameters, ranging over anorder of magnitude from a single bright spot to about 4 mm dimension areobtained. Different driving frequencies ranging from 0.33 to 1.29 MHzare used to generate the annular shapes shown in the figure with fixedamplitude (9.8 V peak to peak) and phase angle chosen to optimize thering shape. For high throughput micromachining, one would use this basisto define a lookup table that establishes the correspondence betweendriving amplitude, frequency, phase shift and the observed instantaneousintensity distribution. A slight eccentricity in the micromachined ringsis noted due to minor imperfections and nonuniformities in fabricatingthe piezoelectric tube 310. Furthermore, these asymmetries depend on thedriving frequency due to resonance behavior in the tube 310. Theconfiguration of the TAG lens 100 of FIGS. 1A, 1B may cancel thisunwanted effect.

Sample Micromachining

The TAG lens 300 is capable of high energy throughput without damage andcan therefore be used for pulsed laser micromachining FIGS. 31A, 31Bdemonstrates this point for a polyimide film. In this case, the incidentlaser energy on the sample is 8.2 μJ and the driving frequency is 700kHz resulting in a 15 μm diameter ring in the film. Calculating theactual fluence is difficult because the background of the beam, althoughsubthreshold, still carries significant energy as expected for aBessel-like beam. A profilometry analysis of the irradiated polyimidethin film shows a well defined annular structure with depth of 0.9 μmand a width of approximately 3.5 μm. Additional studies could beperformed to assess the heat affected zone surrounding the laser-inducedstructures using these non-traditional intensity profiles.

In contrast to many other methods of producing annular beams, the TAGlens 300 gives the added ability to rapidly change the pattern accordingto the structure or pattern required. To demonstrate this effect, FIG.32 gives an example of adjacent laser-induced surface structures thatalternate between a central spot and an annular beam. For this figure,the two basis elements are selected by varying the phase shift such thatthe index function is at either a maximum or minimum while the substrateis manually translated. In the upper image of the figure, a drivingfrequency of 989 kHz is used. The bright central spot shows indicationof a second ring that causes damage to the polyimide film. However, whendriven at lower frequencies as in the lower image (531 kHz), one is ableto obtain single spots in the image. This effect can be ameliorated bybetter optimization of the phase shift and amplitude in order tomaximize the energy difference between the central spot and the outerrings. Furthermore, aperturing or reducing the size of the beam incidenton the TAG lens 300 can be used to remove unwanted outer rings.

Switching Time of the TAG Lens

The ability to switch rapidly between two distinct intensitydistributions is a key parameter in evaluating the relevance of a beamshaping strategy for micromachining or laser marking purposes. Whenusing a TAG lens 100, 200, 300, two situations have to be considered.Either the elements of the basis can be reached by driving the TAG lens300 at a single frequency (FIG. 32), or the lens 300 may be operated atdifferent frequencies, as illustrated in FIG. 30.

When all the desired shapes can be generated by using the same lensdriving frequency, the theoretical minimum switching time is given byhalf of the driving signal period. In FIG. 32, the TAG lens 300 wasdriven at a frequency of 989 kHz (upper picture) and 531 kHz (lowerpicture) implying that toggling times between two adjacent patterns aretheoretically as short as 0.5 μs and 0.9 μs, respectively, or twice thedriving frequency. This converts to a switching frequency ofapproximately 1-2 MHz. Although these values are too low for opticalcommunication and switching requirements, these rates are more thansufficient for pulsed laser processing.

In the case that frequency changes are needed, the minimum amount oftime required to switch the pattern is equal to the amount of time ittakes to propagate the sound wave from the piezoelectric to the centerof the lens 300. This is denoted as the TAG lens 300 response time. Theinstantaneous pattern is established at this time, followed by atransient to reach steady state. In the context of pulsed laserprocessing, it is the response time that is the relevant test of lensspeed. As an example, considering the sound velocity in the silicone oilto be about 900 ms⁻¹ and a radius of 3.5 cm for the lens 300, theresponse time is as short as 40 μs. However, by changing temperature orthe refractive filling fluid, the speed of sound can be increased andthe response time can be significantly decreased.

The effects of transients in the output of the TAG lens 300 to reachsteady state using the silicone oil with a viscosity of 0.65 cS arerelatively fast ranging from 2-3 ms is shown in FIG. 39. The firstpicture in the upper left hand corner shows the TAG output when thedriving frequency is first turned on. Within the first 0.5 ms, thepattern begins to establish a bright central spot with emerging majorrings. These rings continue to sharpen over the next 0.5 ms as does thecentral spot. After 1 ins, the minor rings begin to form and continuedeveloping for the next 0.5 ms. Finally after 2.0 ms, the patternremains steady.

To gain more precise information about the time needed to reach steadystate, a high speed photodiode was used to measure the intensity of thecentral, spot. FIG. 40 illustrates this measurement as the lens 300 isturned off and the acoustic wave dissipates in glycerol. The TAG lens'sdriving voltage is set to zero at t=0. The liquid in the TAG lens 300was replaced with glycerol although results with the silicone oils werealso obtained. The plots are similar but, due to viscosity differences,have different time scales. At negative times, the lens 300 is operatingin the steady state producing the oscillatory behavior discussed inSection I. Once the driving frequency is shut off, the oscillationsrapidly decay toward the constant value. The decay time is quantified byextracting the time for the voltage to decrease to 1/e of the initialsteady state. FIG. 41 shows these results for different filling liquidsand different initial driving frequencies.

The steady state time is expected to be dependent on the viscosity ofthe fluid and the driving frequency. The data in FIG. 41 show a modestdependence on driving frequency that is within the measurement noise. Ahigher viscosity fluid damps out transients and reaches steady statemore quickly as shown in the figure. The average time constant is 2.1ms, 650 μs, and 320 μs for 0.65 cS silicone oil, 100 cS silicone oil,and glycerol respectively. The images of FIG. 39. showing silicone oilat 0.65 cS agree with the decay time measured in this manner. Thereforeone can expect switching rates of 1/decay time or 500-3000 Hz.

V. Dynamic Focusing and Imaging

In another of its aspects, the present invention provides a TAG lens100, 300 and method for a rapidly changing the focal length. The TAGlens 100, 300 is capable of tuning the focal length of converging ordiverging beams by using an aperture to isolate portions of the indexprofile and synchronizing the TAG lens 100, 300 with either a pulsedillumination source, or a pulsed imaging device (camera).

The experimental setup is similar to that described earlier in FIG. 29.In this experiment, a pulsed UV laser 560 is used at 355 nm with 15 nspulse durations and a maximum of 250 kHz repetition rate to illuminatean object of interest 510, FIG. 35. In order to improve imaging andreduce interference, a diffuser or scattering plate, SP, is used. Thelight may be directed using a mirror 512 and beam splitter 514 throughthe object of interest 510 which for this example is a US Air Forcecalibration standard (USAF 1951) that is located at a position, d, awayfrom the entrance to the TAG lens 300. An aperture 520, or iris, withapproximately 1.5 mm interior diameter, is co-axially located on eitherthe input or output side of the TAG lens 300 to restrict optical accessto the portion of the TAG lens 300 at which the desired index ofrefraction is located (cf. FIGS. 21A, 21B, 34 showing refractive indexversus lens radius). Finally a lens, L1, may be disposed in the outputpath of the TAG lens 300 after the aperture 520 to produce an image atan image plane 530 where a detector 540, e.g. CCD camera, is located torecord the image. The filling materials, details of the electroniccircuitry, and the typical driving frequencies for the TAG lens 300 aregiven in paragraph[00204] above.

In order to successfully use the TAG lens 300 as a dynamic focusing andimaging device, it is necessary to synchronize the incident laser pulse(or camera shutter) to trigger at the appropriate temporal phaselocation of the TAG lens driving signal. This is accomplished anddescribed in detail above by using a pulse delay generator that istriggered from the RF signal driving the TAG lens 300. It is possible toaccurately control the exact phase difference between the laser and theTAG driving signals and therefore, the instantaneous state of the indexof refraction profile when light is passing through it.

The detailed physics of the lens operation is described earlier insection I and with the driving voltage at 334 kHz and 9.8 V_(p-p).However, in order to understand this implementation of the TAG lens 300,it is instructive to refer to FIG. 21. As can be seen, at a time t=0(FIG. 21A), the index of refraction profile is at a global maximum inthe center of the lens 300 (ρ or ξ=0) and at half a period later, t=T/2(FIG. 21B), the index of refraction is at a global minimum in thecenter. Based on the sinusoidal temporal dependence of equation for theindex of refraction profile Eq. 58, one can easily see that theinstantaneous profile will oscillate continuously between this globalmaximum and global minimum at ρ=0. At times in between, the indexprofile will be either a local maximum or minimum at ρ=0.

As noted in FIGS. 21A, 21B, if one were to look at the inflection pointsin the index profile, one can approximate the index as linear andproduce a Bessel beam. However, if instead, one were to look only in thesmall region near ρ=0 this region of the index profile is moreaccurately represented as a parabolic function. This result can be seenby taking a Taylor expansion of a Bessel function about ρ=0 and keepingthe lowest order term in ρ which for a Bessel function is ρ². Therefore,if one were to use an aperture to filter out the region of the lens 300that is not within this parabolic region, one is left with an index ofrefraction profile that resembles a simple converging (or diverging)lens 300 with a focal length given by the amplitude and width of thisregion.

What is notable about this interpretation of the TAG lens index ofrefraction profile is that since the curvature, and therefore theeffective focal length, depends on the instantaneous amplitude anddriving frequency of the acoustic wave within the lens 300, theeffective focal length will change continuously in time. Thus in thesame manner that one can synchronize individual patterns of the TAG lens300, one can synchronize the light source, e.g. laser 560, or imagingdevice to select any focal length that is needed subject to thelimitations of the driving signal. For example, synchronizing the lightsource to the pattern in FIG. 21A would give the shortest convergingfocal length while synchronizing to the pattern in FIG. 21B would givethe shortest diverging (negative) focal length.

In order to demonstrate this point, the described apparatus of FIG. 35has been used to image a calibration standard 510 at various locationsalong the optical path. FIG. 36 shows three images of the resolutionstandard 510 located at positions, d=7.5, 26, and 59 cm, from theentrance to the TAG lens 300. As can be seen from the images, at eachlocation, it is possible to accurately generate an image at the imageplane 350 with the CCD array of the camera is located, without changingthe position of the lens 300 relative to the image plane 530. Each ofthe three images in FIG. 35 is taken with a different temporal phasedelay between the TAG and the light source, as illustrated in the graphof FIG. 36 showing the laser pulse timing against TAG driving signal.This result indicates that it is possible to synchronize the lens 300 sothat an object 510 located an arbitrary distance away can be imaged.

In carefully looking at the images in FIG. 36, one notices that not onlyare the images brought into focus at the different locations, but thatthe magnification of the image changes as a function of the positionalong the optical axis as -would be expected. Such an effect isconsistent with the fact that the focal length of a single lens 300 isbeing changed and is a benefit for those desiring not only the abilityto image an object 510 at an arbitrary distance, but also to change thesize of the image. However, one can remove the change in magnificationwith the appropriate combination of standard optical elements andadditional, synchronized TAG lenses in the optical path, if desired.

The speed at which one can move between the different object locationsis exceedingly fast compared to any other adaptive optical element sinceone only needs to change the phase difference between the TAG lens 300and laser driving signal. Therefore, times that are mere fractions ofthe oscillation period can accommodate large changes in the location ofthe object plane. For instance, in FIG. 21A, the amount of time neededto switch between the first image at d=0 and the other two images is0.28 and 0.48 microseconds, respectively. A more complete discussion onthe switching time is presented in section IV above.

In addition to being able to move an object and image it at a differentlocation, it is possible to rapidly switch between existing objectslocated at different places on the optical axis. In FIG. 37, anexperimental setup similar to that of FIG. 35 is shown, but instead of acalibration standard 510, there is a series of wires A, B, C positionedat different locations in front of the TAG lens 300 with differentoffset angles to render each wire A, B, C identifiable in the imageplane 530. In this setup, the lens 300 can be synchronized with thelaser pulse to bring any one of the three wires A, B, C, into focus onthe image plane 530 without the need to remove the other two non-imagedwires. This also indicates that the apertured TAG lens 300 is not justacting as a pinhole camera because only one wire A, B, C is in focus inany given image.

The other notable aspect of this experiment is that the TAG lens 300 canbe synchronized to be either a converging or diverging lens 300depending upon the phase difference. In the experimental setup with thethree wires A, B, C, the focal length of the lens L1 and the distanceson the optical axis are configured so that it is necessary for the TAGlens 300 to be either converging, diverging, or planar in order to bringone of the wires A, B, C into focus. As can be seen from the image inFIG. 38A, when the laser 560 is synchronized so that it illuminates theTAG lens 300 while the instantaneous index of refraction shows negativecurvature, a converging lens 300 is encountered and the wire A closestto the lens 300 is brought into focus. When the laser is synchronized topass through the TAG lens 300 when the index function has zeroamplitude, the TAG lens 300 acts as a flat plate and therefore does notdisrupt the wavefront curvature. Thus, the middle wire B is in focus byoperation of the lens L1 alone, FIG. 38B. Finally, when thesynchronization is done such that the instantaneous index of refractionexhibits positive curvature and the laser pulse passes through the lens300, it behaves as a diverging lens 300 and the furthest wire C isbrought into focus, FIG. 38C. Thus, ability to dynamically position thefocal position of a converging or diverging TAG lens 300, or to rapidlychange the location of and magnification in an image plane, enablesfundamentally new methods of imaging with great potential in the fieldsof industrial controls, homeland security, biological imaging and manyother important areas.

These and other advantages of the present invention will be apparent tothose skilled in the art from the foregoing specification. Accordingly,it will be recognized by those skilled in the art that changes ormodifications may be made to the above-described embodiments withoutdeparting from the broad inventive concepts of the invention. It shouldtherefore be understood that this invention is not limited to theparticular embodiments described herein, but is intended to include allchanges and modifications that are within the scope and spirit of theinvention as set forth in the claims.

What is claimed is:
 1. A method for driving a tunable acoustic gradientindex of refraction lens to produce a desired refractive indexdistribution within the lens, comprising: selecting a desired refractiveindex distribution to be produced within the lens; determining thefrequency response of the lens; using the frequency response todetermine a transfer function of the lens to relate the index responseto voltage input; decomposing the desired refractive index distributioninto its spatial frequencies; \ converting the spatial frequencies intotemporal frequencies; representing the voltage input as an expansionhaving voltage coefficients; determining the voltage coefficients fromthe representation of the decomposed refractive index distribution;using the determined voltage coefficients to determine the voltage inputin the time domain; and driving a tunable acoustic gradient index ofrefraction lens with the determined voltage input responsively to acontrol signal.
 2. The method according to claim 1, wherein the step ofconverting the spatial frequencies comprises converting the decomposedrefractive index distribution into discrete spatial frequencies toprovide a discretized representation of the decomposed refractive indexdistribution.
 3. The method according to claim 1, wherein the step ofdetermining the voltage coefficients is preformed before the step ofrepresenting the voltage input.
 4. The method according to claim 1,wherein the step of decomposing a desired refractive index distributioncomprises using a Fourier-Bessel transform to decompose the refractiveindex into its spatial frequencies.
 5. The method according to claim 1,wherein the step of determining the voltage coefficients comprises usingan inverse Fourier-Bessel transform.
 6. The method according to claim 1,wherein the step of decomposing a desired refractive index distributioncomprises using a Fourier transform to decompose the refractive indexinto its spatial frequencies.
 7. The method according to claim 1,wherein the step of determining the voltage coefficients comprises usingan inverse Fourier transform.
 8. The method according to claim 1,wherein the step of converting the spatial frequencies, comprisesconverting into discrete temporal frequencies that are integer multiplesof the ratio of a frequency with which the desired refractive indexdistribution repeats in time within the lens to the speed of soundwithin the lens.
 9. The method according to claim 1, wherein the desiredrefractive index distribution comprises a parabolic refractive indexdistribution, where the refractive index in the lens varies as thesquare of the radius of the lens.